The following problem is exercise 170 in Golan's linear algebra book. I have posted a solution attempt in the answers.
Problem: Let $V$, $W$, and $Y$ be vector spaces finitely-generated over a field $F$ and let $\alpha:V\rightarrow W$ be a linear transformation. Show that the set of all linear transformations $\beta: W\rightarrow Y$ satisfying the condition that $\beta\circ \alpha$ is the 0-transformation is a subspace of $\operatorname{Hom}(W,Y)$, and compute its dimension.