IMC / 1998 / Problems / Day 1, P1

IMC 1998 · Day 1 · P1

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.

Solution (official)

First choose a basis $\{v_1, v_2, v_3\}$ of $U_1$ . It is possible to extend this basis with vectors $v_4, v_5$ and $v_6$ to get a basis of $U_2$ . In the same way we can extend a basis of $U_2$ with vectors $v_7, \dots, v_{10}$ to get as basis of $V$ .

Let $T \in \mathcal{E}$ be an endomorphism which has $U_1$ and $U_2$ as invariant subspaces. Then its matrix, relative to the basis $\{v_1, \dots, v_{10}\}$ is of the form $\begin{pmatrix} * & * & * & * & * & * & * & * & * & * \\ * & * & * & * & * & * & * & * & * & * \\ * & * & * & * & * & * & * & * & * & * \\ 0 & 0 & 0 & * & * & * & * & * & * & * \\ 0 & 0 & 0 & * & * & * & * & * & * & * \\ 0 & 0 & 0 & * & * & * & * & * & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & * & * & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & * & * & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & * & * & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & * & * & * & * \end{pmatrix}.$ So $\dim_{\mathbb{R}} \mathcal{E} = 9 + 18 + 40 = 67$ .

IMC 1998 · Day 1 · P1

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