|
: ''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 (linear algebra)」の詳細全文を読む スポンサード リンク
|