grandes-ecoles 2018 Q36

grandes-ecoles · France · centrale-maths1__official Proof Deduction or Consequence from Prior Results

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the Frobenius norm $\|\cdot\|_{F}$. We fix a unit vector $u$ in $\mathbb{R}^{d}$, define $g(M) = \|M \cdot u\|$, and let $C = \{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\}$. Let $r$ and $t$ be two real numbers, with $t > 0$. Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$
$$d(M, C) < t \quad \Longrightarrow \quad g(M) < r + t$$

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the Frobenius norm $\|\cdot\|_{F}$. We fix a unit vector $u$ in $\mathbb{R}^{d}$, define $g(M) = \|M \cdot u\|$, and let $C = \{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\}$. Let $r$ and $t$ be two real numbers, with $t > 0$. Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$

$$d(M, C) < t \quad \Longrightarrow \quad g(M) < r + t$$

Paper Questions

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