Linear mapping/Change of basis/No proof/Section

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Let denote a field, and let and denote finite-dimensional -vector spaces. Let and be bases of and and bases of . Let

denote a linear mapping, which is described by the matrix with respect to the bases and . Then is described with respect to the bases and by the matrix

where and are the transformation matrices, which describe the change of basis from to and from to .


This proof was not presented in the lecture.

Let denote a field, and let denote a -vector space of finite dimension. Let

be a linear mapping. Let and denote bases of . Then the matrices which describe the linear mapping with respect to and respectively (on both sides), fulfil the relation

This follows directly from fact.

Two square matrices are called similar, if there exists an invertible matrix with


Due to fact, for a linear mapping , the describing matrices with respect to several bases are similar.