Linear mapping/Matrix/Commutative diagram/Fact

Let ${}K$ denote a field and let ${}V$ denote an ${}n$-dimensional vector space with a basis ${}{\mathfrak {v}}=v_{1},\ldots ,v_{n}$. Let ${}W$ be an ${}m$-dimensional vector space with a basis ${}{\mathfrak {w}}=w_{1},\ldots ,w_{m}$, and let

$\Psi _{\mathfrak {v}}\colon K^{n}\longrightarrow V$

and

$\Psi _{\mathfrak {w}}\colon K^{m}\longrightarrow W$

be the corresponding mappings. Let

$\varphi \colon V\longrightarrow W$

denote a linear mapping with describing matrix ${}M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )$.

Then

${}\varphi \circ \Psi _{\mathfrak {v}}=\Psi _{\mathfrak {w}}\circ M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )\,$

hold, that is, the diagram

${\begin{matrix}K^{n}&{\stackrel {\Psi _{\mathfrak {v}}}{\longrightarrow }}&V&\\\!\!\!\!\!M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )\downarrow &&\downarrow \varphi \!\!\!\!\!&\\K^{m}&{\stackrel {\Psi _{\mathfrak {w}}}{\longrightarrow }}&W&\!\!\!\!\!\\\end{matrix}}$

commutes.

For a vector ${}v\in V$, we can compute ${}\varphi (v)$ by determining the coefficient tuple of ${}v$ with respect to the basis ${}{\mathfrak {v}}$, applying the matrix ${}M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )$ and determining for the resulting ${}m$-tuple the corresponding vector with respect to ${}{\mathfrak {w}}$.

Write a proof