Endomorphismus/Finite dimensional/Change of basis/Fact

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$ -vector space of finite dimension. Let

\varphi \colon V\longrightarrow V

be a linear mapping. Let ${}{\mathfrak {u}}$ and ${}{\mathfrak {v}}$ denote bases of ${}V$ .

Then the matrices that describe the linear mapping with respect to

${}{\mathfrak {u}}$ and ${}{\mathfrak {v}}$ respectively (on both sides), fulfil the relation

{}M_{\mathfrak {u}}^{\mathfrak {u}}(\varphi )=M_{\mathfrak {u}}^{\mathfrak {v}}\circ M_{\mathfrak {v}}^{\mathfrak {v}}(\varphi )\circ (M_{\mathfrak {u}}^{\mathfrak {v}})^{-1}\,.