Homomorphism space/Bases/Matrices/Fact

Let $K$ denote a field and let $V$ and $W$ denote finite-dimensional $K$-vector spaces. Let $𝔳=v_1,\dots ,v_n$ be a basis of $V$ and $𝔴=w_1,\dots ,w_m$ be a basis of $W$.

Then the assignment

${\mathrm{Hom}}_K(V,W)⟶{\mathrm{Mat}}_{m\times n}(K),f⟼M_{𝔴}^{𝔳}(f),$

is an isomorphism

of $K$-vector spaces.

Proof, Write another proof