grandes-ecoles 2020 Q21

grandes-ecoles · France · centrale-maths1__pc Matrices Projection and Orthogonality

Let $Y_{1}, \ldots, Y_{p}$ be vectors of $\mathcal{M}_{2n,1}(\mathbb{R})$. Let $Y \in \mathcal{M}_{2n,1}(\mathbb{R})$. Show the implication $$Y \in \left(\operatorname{Vect}(Y_{1}, \ldots, Y_{p}, J_{n} Y_{1}, \ldots, J_{n} Y_{p})\right)^{\perp} \Longrightarrow J_{n} Y \in \left(\operatorname{Vect}(Y_{1}, \ldots, Y_{p}, Y, J_{n} Y_{1}, \ldots, J_{n} Y_{p})\right)^{\perp}.$$

Let $Y_{1}, \ldots, Y_{p}$ be vectors of $\mathcal{M}_{2n,1}(\mathbb{R})$. Let $Y \in \mathcal{M}_{2n,1}(\mathbb{R})$. Show the implication
$$Y \in \left(\operatorname{Vect}(Y_{1}, \ldots, Y_{p}, J_{n} Y_{1}, \ldots, J_{n} Y_{p})\right)^{\perp} \Longrightarrow J_{n} Y \in \left(\operatorname{Vect}(Y_{1}, \ldots, Y_{p}, Y, J_{n} Y_{1}, \ldots, J_{n} Y_{p})\right)^{\perp}.$$

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