Linear mapping/Matrix/Composition/Fact

More precisely: let ${\displaystyle {}U,V,W}$ denote vector spaces over a field ${\displaystyle {}K}$, with bases

${\displaystyle {\mathfrak {u}}=u_{1},\ldots ,u_{p},\,{\mathfrak {v}}=v_{1},\ldots ,v_{n},{\text{ and }}{\mathfrak {w}}=w_{1},\ldots ,w_{m}.}$

Let

${\displaystyle \psi :U\longrightarrow V{\text{ and }}\varphi :V\longrightarrow W}$

denote linear mappings. Then, for the describing matrix of ${\displaystyle {}\psi ,\,\varphi }$, and of the composition ${\displaystyle {}\varphi \circ \psi }$, the relation

${\displaystyle {}M_{\mathfrak {w}}^{\mathfrak {u}}(\varphi \circ \psi )=(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ))\circ (M_{\mathfrak {v}}^{\mathfrak {u}}(\psi ))\,}$

holds.