Let $F$ be a vector subspace of $E$ of dimension $p$ and $\mathcal{A}_{F}$ the set of endomorphisms of $E$ that stabilise $F$, that is $\mathcal{A}_{F} = \{ u \in \mathcal{L}(E) \mid u(F) \subset F \}$.
Show that $\mathcal{A}_{F}$ is a subalgebra of $\mathcal{L}(E)$.