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Let $L: V \rightarrow V$ be a linear map on a vector space $V$.
a) Show that $\operatorname{ker} L \subset \operatorname{ker} L^{2}$ and, more generally, $\operatorname{ker} L^{k} \subset \operatorname{ker} L^{k+1}$ for all $k \geq 1$.
b) If $\operatorname{ker} L^{j}=\operatorname{ker} L^{j+1}$ for some integer $j$, show that $\operatorname{ker} L^{k}=\operatorname{ker} L^{k+1}$ for all $k \geq j$. Does your proof require that $V$ is finite dimensional?
c) Let $A$ be an $n \times n$ matrix. If $A^{j}=0$ for some integer $j$ (perhaps $j>n$ ), show that $A^{n}=0$.
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