IMC 1998 · Problem 1

Let $V$ be a 10-dimensional real vector space and $U_1$ and $U_2$ two linear subspaces such that $U_1 \subseteq U_2$, $\dim_{\mathbb{R}} U_1 = 3$ and $\dim_{\mathbb{R}} U_2 = 6$. Let $\mathcal{E}$ be the set of all linear maps $T : V \longrightarrow V$ which have $U_1$ and $U_2$ as invariant subspaces (i.e., $T(U_1) \subseteq U_1$ and $T(U_2) \subseteq U_2$). Calculate the dimension of $\mathcal{E}$ as a real vector space.