# Linear mapping/Change of basis/No proof/Section

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 .

### Proof

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.