Linear mapping/Matrix/Commutative diagram/Fact

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

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

and

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

be the corresponding mappings. Let

${\displaystyle \varphi \colon V\longrightarrow W}$

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

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