Homomorphism space/Bases/Matrices/Fact

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

Then the assignment

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

is an isomorphism

of ${}K$-vector spaces.

Proof, Write another proof