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 $u$ is an orthonormal basis of $V$. (b) Show that for $k \in \llbracket 1, p-1 \rrbracket$, we have $u_k^{\prime} \in \operatorname{Vect}(u_{k+1}, \ldots, u_p)^{\perp}$. (Hint: One may consider the map $t \mapsto u_k(t) = \frac{u_k + t u_l}{\|u_k + t u_l\|}$ for all $t \in \mathbb{R}$ and $l \in \llbracket k+1, p \rrbracket$ as well as its derivative.) (c) Show that $u_{k+1} \in \left(\operatorname{Vect}(u_1, \ldots, u_k) + \operatorname{Vect}(u_1^{\prime}, \ldots, u_k^{\prime})\right)^{\perp}$ for all $k$ element of $\llbracket 1, p-1 \rrbracket$. (d) Deduce that the subspaces $W_k = \operatorname{Vect}(u_k, u_k^{\prime})$ for $k \in \llbracket 1, p \rrbracket$ are pairwise orthogonal.
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 $u$ is an orthonormal basis of $V$.
(b) Show that for $k \in \llbracket 1, p-1 \rrbracket$, we have $u_k^{\prime} \in \operatorname{Vect}(u_{k+1}, \ldots, u_p)^{\perp}$.
(Hint: One may consider the map $t \mapsto u_k(t) = \frac{u_k + t u_l}{\|u_k + t u_l\|}$ for all $t \in \mathbb{R}$ and $l \in \llbracket k+1, p \rrbracket$ as well as its derivative.)
(c) Show that $u_{k+1} \in \left(\operatorname{Vect}(u_1, \ldots, u_k) + \operatorname{Vect}(u_1^{\prime}, \ldots, u_k^{\prime})\right)^{\perp}$ for all $k$ element of $\llbracket 1, p-1 \rrbracket$.
(d) Deduce that the subspaces $W_k = \operatorname{Vect}(u_k, u_k^{\prime})$ for $k \in \llbracket 1, p \rrbracket$ are pairwise orthogonal.