翻訳と辞書 Words near each other ・ Basire ・ Basireddipalle ・ Basireddypally ・ Basirepomys ・ Basirhat ・ Basirhat (Lok Sabha constituency) ・ Basirhat College ・ Basirhat Dakshin (Vidhan Sabha constituency) ・ Basirhat I (community development block) ・ Basirhat II (community development block) ・ Basirhat subdivision ・ Basirhat Uttar (Vidhan Sabha constituency) ・ Basiria ・ Basirpur railway station ・ Basis ・ Basis (linear algebra) ・ Basis (universal algebra) ・ Basis database ・ Basis function ・ BASIS Independent Silicon Valley ・ BASIS International ・ Basis Nord ・ Basis of accounting ・ Basis of articulation ・ Basis of estimate ・ Basis of Union ・ Basis of Union (Presbyterian Church of Australia) ・ Basis of Union (Uniting Church in Australia) ・ Basis path testing ・ Basis point Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Basis (linear algebra) ： ウィキペディア英語版 : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a'n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F, if ''a''1''v''1 + … + ''a'n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' = 0, then necessarily ''a''1 = … = ''a'n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a'n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a'n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a'n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F and ''v''1, …, ''v'n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a'n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. : ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.'' A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.〔Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4〕 In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space. == Definition == A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''. In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions: * the ''linear independence'' property, :: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and * the ''spanning'' property, :: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if * every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and * for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F, if ''a''1''v''1 + … + ''a'n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' = 0, then necessarily ''a''1 = … = ''a'n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a'n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a'n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a'n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F and ''v''1, …, ''v'n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a'n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F, if ''a''1''v''1 + … + ''a'n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' = 0, then necessarily ''a''1 = … = ''a'n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a'n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a'n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a'n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F and ''v''1, …, ''v'n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a'n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a '''basis''', or a set of '''basis vectors''', if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector '''coordinates''' or '''components'''. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A '''basis''' ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field '''F''' (such as the real or complex numbers '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a''''n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.">ウィキペディアで「: ''Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.''A set of vectors in a vector space ''V'' is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, (page 10 ), ISBN 0-387-90093-4 In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space ''V'', every element of ''V'' can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.== Definition ==A basis ''B'' of a vector space ''V'' over a field ''F'' is a linearly independent subset of ''V'' that spans ''V''.In more detail, suppose that ''B'' = is a finite subset of a vector space ''V'' over a field F (such as the real or complex numbers R or C). Then ''B'' is a basis if it satisfies the following conditions:* the ''linear independence'' property,:: for all ''a''1, …, ''a'n'' ∈ '''F''', if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F, if ''a''1''v''1 + … + ''a'n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' = 0, then necessarily ''a''1 = … = ''a'n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' = 0; and* the ''spanning'' property,:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a'n'' ∈ '''F''' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a'n''''v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n''.The numbers ''a''i are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) ''B'' ⊂ ''V'' is a basis, if* every finite subset ''B''0 ⊆ ''B'' obeys the independence property shown above; and* for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a'n'' ∈ '''F''' and ''v''1, …, ''v''''n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ F and ''v''1, …, ''v'n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'' ∈ ''B'' such that ''x'' = ''a''1''v''1 + … + ''a'n''''v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n'v''''n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む v'n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an '''ordered basis''', which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む n''.The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions'' below.It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span ''V'': see ''Ordered bases and coordinates'' below.」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.