Bilinear form dimension

Question

I saw that the dimension of a Bilinear form is: $$ \dim B(V,W) = \dim V \cdot \dim W $$

where a bilinear form means (in my course): $$ B : V\times W \rightarrow F $$ I don't really understand why, and I can't find a proper explanation anywhere.

Answer 1

Let $(e_1,\ldots,e_n)$ and $(f_1,\ldots,f_m)$ a basis of $V$ and $W$ respectively and let

$$\Phi\colon B(V,W)\rightarrow \mathcal M_{nm}(\Bbb R), \varphi\mapsto \left(\varphi(e_i,f_j)\right)_{ij}$$ so we can see that $\Phi$ is an isomorphism of linear vector spaces so $$\dim B(V,W)=\dim \mathcal M_{nm}(\Bbb R)=nm=\dim V\times \dim W$$