grandes-ecoles 2017 Q2

grandes-ecoles · France · x-ens-maths__psi Matrices Linear Transformation and Endomorphism Properties

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Show that for all $k \in \{0, 1, \ldots, 2m\}$, $\operatorname{Im}(T^{k+1}) \subset \operatorname{Im}(T^k)$ and $\operatorname{Im}(T^{k+1}) \neq \operatorname{Im}(T^k)$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.

Show that for all $k \in \{0, 1, \ldots, 2m\}$, $\operatorname{Im}(T^{k+1}) \subset \operatorname{Im}(T^k)$ and $\operatorname{Im}(T^{k+1}) \neq \operatorname{Im}(T^k)$.

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