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 \(A^{n}=0\).