another question from a problem set that I am struggling to answer. Any guidance or help would be appreciated. I don't even really understand what the question is asking.
Let $V$ be a nonzero vector space and let $T \in \text{Hom}(V, V )$. Show that $\{S \in Hom(V, V ) : S \circ T = T \circ S \}$ is a subspace of $ \text{Hom}(V, V )$.
Let us call the given set K. You will be done if you can show the following two steps:
If $S_1,S_2 \in K$, then $S_1 + S_2 \in K$.
If $S \in K$, then $kS \in K$ for all $k$.
To show the first, note that $(S_1+S_2) \circ T = S_1 \circ T + S_2 \circ T = T \circ S_1 + T \circ S_2 = T \circ (S_1 + S_2)$.
The second simply follows: $(kS) \circ T = k(S \circ T) = k(T \circ S) = T \circ (kS)$.
You can verify these steps yourself, but do ask if doubts persist. The above two steps show that $K$ is closed under addition, and scalar multiplication respectively, so $K$ is a subspace of $\operatorname{Hom} (V,V)$.
ADDENDUM : I should mention, thanks to the comment below, that $K$ is non-empty , since $T$ commutes with itself, hence is in $K$. (The identity map is in $K$ as well, as is the zero map!)