This is from Axler's Linear Algebra Done Right:
3.20 Proposition
If $V$ and $W$ are finite dimensional, then $L(V , W )$ is finite dimensional and
dim $L(V,W)=(\dim V)(\dim W)$.
Proof:
This follows from the equation $\dim \,\operatorname{Mat}(m, n, F) = mn$, 3.18, and 3.19.
where 3.18 states:
Two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension.
and 3.19 states:
Suppose that $(v_1 , \ldots , v_n)$ is a basis of $V$ and $(w_1, \ldots ,w_m)$ is a basis of $W$. Then $M$ is an invertible linear map between $L(V , W )$ and $\operatorname{Mat}(m, n, F)$.
Can someone explain the proof of 3.20 more clearly because I do not really follow.
Thanks