Homomorphism space/Bases/Matrices/Fact

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

Then the assignment

${\displaystyle \operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Mat} _{m\times n}(K),f\longmapsto M_{\mathfrak {w}}^{\mathfrak {v}}(f),}$

is an isomorphism

of ${\displaystyle {}K}$-vector spaces.