Let $V$ be an $n$-dimensional complex vector space, and let $T: V \to V$ be a linear transformation.
(i) For $i > 0$, let $K_i$ = $\ker T^i$($T_0$ means the identity transformation). Show that for each $i$, $K_i ⊆ K_{i+1}$, and deduce that there exists a non-negative integer $r$ such that $K_r = K_{r+1}$. Prove that $K_r = K_{r+j}$ for all $j \geq 1$. Hence, or otherwise, show that $V = K_r ⊕ T^r(V )$.
(ii) Suppose that the only eigenvalues of $T$ are $0$ and $λ$, where $λ\neq0$.Let $W = T^r(V)$, where r is as above. Show that $T(W) ⊆ W$, and that the restriction of $T$ to $W$ has $λ$ as its only eigenvalue. Let $S$ denote the restriction of $(T − λI)$ to $W$. Show that $0$ is the only eigenvalue of S. By applying (i) with $S$, $W$ in place of $T$, $V$ , show that $S^m(W) = 0$ for some $m$.
for (ii), how to show $T(W) ⊆ W$ and the restriction of $T$ to $W$ has only $\lambda$?
1) you have $T(V)\subset V$.What do you obtain if you apply $T^r$ ?
2) to rule out $0$ as an eigenvalue, you need to show that $\ker(T_{\vert_W})=0.$
If $w=T^r(v)\in W$ satisfies $T(w)=0$, what can you say about $v$ ? Then use the definition of $r$ and i).