Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$. Let $u = (u_1, \ldots, u_p)$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ be the orthonormal families constructed in question (1). (a) Show that there exist $0 \leqslant \theta_1 \leqslant \cdots \leqslant \theta_p \leqslant \pi/2$ such that $\cos(\theta_k) = \langle u_k, u_k^{\prime}\rangle$ for all $k \in \llbracket 1, p \rrbracket$. (b) Calculate the value of $\operatorname{det}(\operatorname{Gram}(u, u^{\prime}))$ as a function of the $\cos(\theta_k)$. (c) Deduce that $\operatorname{det}(\operatorname{Gram}(u, u^{\prime})) \leqslant 1$. What can be said about $V$ and $V^{\prime}$ in the case of equality?
Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$. Let $u = (u_1, \ldots, u_p)$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ be the orthonormal families constructed in question (1).
(a) Show that there exist $0 \leqslant \theta_1 \leqslant \cdots \leqslant \theta_p \leqslant \pi/2$ such that $\cos(\theta_k) = \langle u_k, u_k^{\prime}\rangle$ for all $k \in \llbracket 1, p \rrbracket$.
(b) Calculate the value of $\operatorname{det}(\operatorname{Gram}(u, u^{\prime}))$ as a function of the $\cos(\theta_k)$.
(c) Deduce that $\operatorname{det}(\operatorname{Gram}(u, u^{\prime})) \leqslant 1$. What can be said about $V$ and $V^{\prime}$ in the case of equality?