prove that the Dimension for set of all linear mappings $L:V \to W$ is equal to $\dim V \times \dim W$

Question

I wish to prove that the dimension for the set of all linear mappings $L:V \to W$ is equal to $\dim V \times \dim W$. I know that any general linear mapping can be represented as a matrix, so intuitevely it makes sense that the dimension should be $\dim V \times \dim W$ but I cannot find a way to prove this mathematically

Answer 1

Let $m=\dim V, n=\dim W$. Any linear map $$L:\Bbb R^m\rightarrow\Bbb R^n$$ may be represented as an $n\times m$-matrix $$\mathbf{L}=\begin{pmatrix}a_{1,1}&a_{1,2}&\dots&a_{1,m}\\a_{2,1}&a_{2,2}&\dots&a_{2,m}\\\vdots&\vdots&&\vdots\\a_{n,1}&a_{n,2}&\dots&a_{n,m}\end{pmatrix}$$

How many different such $\mathbf{L}$ can you make?