grandes-ecoles 2017 Q4

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 $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Deduce also that $\operatorname{Im}(T^k) = \operatorname{ker}(T^{2m+1-k})$ for $0 \leq k \leq 2m+1$.

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 $T^{2m+1} = 0_{\mathcal{L}(E)}$.

Deduce also that $\operatorname{Im}(T^k) = \operatorname{ker}(T^{2m+1-k})$ for $0 \leq k \leq 2m+1$.

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