I am very confused about the following exercise:
Let $V, W$ be vector spaces over a field $F$. Show that $Hom_F(V, W)$ is a vector subspace of the set of all mappings $Maps(V, W)$ from $V$ to $W$.
It is intuitively clear why the set of all $Hom_F(V, W)$ would be a subset of all mappings $V\to W$, however, I am confused about how a linear mapping can be a vector subspace at all.
Furthermore, the exercise then asks: Show that: $$dim(Hom_F(V, W))= (dim V )(dim W)$$
How can a homomorphism have a dimension in the same way as a vector space? I cannot conceptualize this. I know a linear mapping $Hom(,)$ is isomorphic to $Mat(,)$, the set of matrices ×, where $=dim$ and $=dim$, thus the dimension can be found this way. But my solution set refers to letting $f_{ij} : V \to W$ and claiming that $f_{ij}$ is linear, and also that ${\{f_{ij}\}}_{i\in I,j\in J}$ is linearly independent. I cannot see how this is related to the above question at all.
Looking for someone to clear up everything going on here. Thanks!