Linear mapping/Matrix/Composition/Fact

In the

correspondence between linear mappings and matrices, the composition of linear mappings corresponds to the

matrix multiplication.

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

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

Let

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

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

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

holds.

Proof 1, 2, Write another proof