Let $V,W$ be vector spaces with finite dimension over a field $K$ , let $A\subseteq V, B\subseteq W$ be vector subspaces. Denote $\hom(V,W)$ the vector space of all linear applications from $V$ to $W$, and define:$$H:=\lbrace f\in \hom(V,W)\;|\;f(A)\subseteq B \rbrace$$ Show that $H$ is a vector subspace of $\hom(V,W)$ and that the dimension of $H$ is $(\dim(V)-\dim(A))\dim(W)+\dim(A)\dim(B)$
Tip: Choose a base of $A$ and $B$, extend it to a base of $V$ and $W$, What form do matrices rappresenting an $f \in H$ with respect to those basis have?
This is a problem given on my textbook, I've shown that $H$ is a vector subspace with just a few trivial calculations, I'm not sure how to find the dimension of $H$, I'm pretty sure I have to utilize the fact that there exist an isomorphism between linear applications and matrices thus having a way esier computation for the dimension (i.e. the number of rows times the number of columns of the matrix), but that being said I'm not sure how to actually arrive at the desired conclusion, any help is appreciated
Hint: Expanding on the tip - suppose that $v_1,\dots,v_j$ forms a basis of $A$, $v_1,\dots,v_n$ forms a basis of $V$, $w_1,\dots,w_k$ forms basis of $B$, and $w_1, \dots, w_m$ forms a basis of $W$. Show that $T:V \to W$ is an element of $H$ if and only if the matrix of $T$ with respect to the bases $\{v_1,\dots,v_n\}$ and $\{w_1,\dots,w_m\}$ has the form $$ M = \pmatrix{A&B\\0&C} $$ where $A$ is $k \times j$ and $M$ is $m \times n$.