We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that $$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$ Let $(e_1, \ldots, e_n)$ be the canonical basis of $\mathbb{R}^n$ and, for $i \in \llbracket 1, n \rrbracket$, let $u_i$ be the endomorphism of $\mathbb{R}^n$ defined by $$\forall X \in \mathbb{R}^n, \quad u_i(X) = B(X, e_i)$$ The matrix of $u_i$ in the canonical basis of $\mathbb{R}^n$ will be denoted $A_i$. a) Prove that, for all $X \in \mathbb{R}^n$, we have $$\forall Y = (y_1, \ldots, y_n) \in \mathbb{R}^n, \quad \sum_{i,j=1}^n y_i y_j (u_i(X) \mid u_j(X)) = \|X\|^2 \sum_{i=1}^n y_i^2$$ b) Deduce that the endomorphisms $u_i$ satisfy the relations $$\forall i, j = 1, \ldots, n, \forall X \in \mathbb{R}^n, \quad \|u_i(X)\| = \|X\| \text{ and } i \neq j \Rightarrow (u_i(X) \mid u_j(X)) = 0$$ and more generally $$\forall i, j = 1, \ldots, n, \forall X, X' \in \mathbb{R}^n, \quad (u_i(X) \mid u_i(X')) = (X \mid X') \text{ and } i \neq j \Rightarrow (u_i(X) \mid u_j(X')) + (u_j(X) \mid u_i(X')) = 0$$ c) Prove that the matrices $A_i$ satisfy the relations $\forall i, j = 1, \ldots, n, \quad {}^t A_i A_i = I_n$ and $i \neq j \Rightarrow {}^t A_i A_j + {}^t A_j A_i = 0$.
We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that
$$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$
Let $(e_1, \ldots, e_n)$ be the canonical basis of $\mathbb{R}^n$ and, for $i \in \llbracket 1, n \rrbracket$, let $u_i$ be the endomorphism of $\mathbb{R}^n$ defined by
$$\forall X \in \mathbb{R}^n, \quad u_i(X) = B(X, e_i)$$
The matrix of $u_i$ in the canonical basis of $\mathbb{R}^n$ will be denoted $A_i$.\\
a) Prove that, for all $X \in \mathbb{R}^n$, we have
$$\forall Y = (y_1, \ldots, y_n) \in \mathbb{R}^n, \quad \sum_{i,j=1}^n y_i y_j (u_i(X) \mid u_j(X)) = \|X\|^2 \sum_{i=1}^n y_i^2$$
b) Deduce that the endomorphisms $u_i$ satisfy the relations
$$\forall i, j = 1, \ldots, n, \forall X \in \mathbb{R}^n, \quad \|u_i(X)\| = \|X\| \text{ and } i \neq j \Rightarrow (u_i(X) \mid u_j(X)) = 0$$
and more generally
$$\forall i, j = 1, \ldots, n, \forall X, X' \in \mathbb{R}^n, \quad (u_i(X) \mid u_i(X')) = (X \mid X') \text{ and } i \neq j \Rightarrow (u_i(X) \mid u_j(X')) + (u_j(X) \mid u_i(X')) = 0$$
c) Prove that the matrices $A_i$ satisfy the relations $\forall i, j = 1, \ldots, n, \quad {}^t A_i A_i = I_n$ and $i \neq j \Rightarrow {}^t A_i A_j + {}^t A_j A_i = 0$.