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In linear algebra, a basis for a vector space of dimension ''n'' is a sequence of ''n'' vectors with the property that every vector in the space can be expressed uniquely as a linear combination of the basis vectors. The matrix representations of operators are also determined by the chosen basis. Since it is often desirable to work with more than one basis for a vector space, it is of fundamental importance in linear algebra to be able to easily transform coordinate-wise representations of vectors and operators taken with respect to one basis to their equivalent representations with respect to another basis. Such a transformation is called a change of basis. Although the terminology of vector spaces is used below and the symbol ''R'' can be taken to mean the field of real numbers, the results discussed hold whenever ''R'' is a commutative ring and ''vector space'' is everywhere replaced with ''free R-module''. == Preliminary notions == The standard basis for ''Rn'' is the ordered sequence , where e''j'' is the element of ''Rn'' with 1 in the ''j''th place and 0s elsewhere. If is a linear transformation, the matrix of ''T'' is the matrix t whose ''j''th column is ''T''(e''j'') for . In this case we have for all x in ''Rn'', where we regard x as a column vector and the multiplication on the right side is matrix multiplication. It is a basic fact in linear algebra that the vector space of all linear transformations from ''Rn'' to ''Rm'' is naturally isomorphic to the space of matrices over ''R''; that is, a linear transformation is for all intents and purposes equivalent to its matrix t. We will also make use of the following simple observation. Theorem Let ''V'' and ''W'' be vector spaces, let be a basis for ''V'', and let be any ''n'' vectors in ''W''. Then there exists a unique linear transformation with for . This unique ''T'' is defined by . Of course, if happens to be a basis for ''W'', then ''T'' is bijective as well as linear; in other words, ''T'' is an isomorphism. If in this case we also have , then ''T'' is said to be an automorphism. Now let ''V'' be a vector space over ''R'' and suppose is a basis for ''V''. By definition, if ξ is a vector in ''V'' then for a unique choice of scalars in ''R'' called the coordinates of ξ relative to the ordered basis The vector in ''Rn'' is called the coordinate tuple of ξ (relative to this basis). The unique linear map with for is called the coordinate isomorphism for ''V'' and the basis Thus if and only if . 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「Change of basis」の詳細全文を読む スポンサード リンク
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