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Let $L: V\to V$ be an operator in a finite-dimensional vector space $V$ over $R$. For any $n \geq 0$, let $K_n = \ker (L^n)$, $I_n = \mathrm{Im}(L^n)$.

(a) How do we prove the stabilization property:there exists $N$ such that for all $n \geq N$, we have $K_n=K_N$ , $I_n= I_N$?

(b) Denote $K = K_N$, $I = I_N$, where $N$ is the same as above. How do we prove that $LK$ is contained in $K$, and $LI$ is contained in $I$, and the restriction of $L$ to $K$ is nilpotent, restriction of $L$ to $I$ is invertible?

(c) How do we prove that $V = K \oplus I$?

For (a), I think we can use the dimension theorem, but how will this work in the proof?

We can assume without proof that if $p \in R[x]$ is the characteristic polynomial of $L$, then $p(L) = 0$ , but, ?how to proceed?

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(a) Can you show that $K_1\subseteq K_2$ and that $I_2\subseteq I_1$? if so, conclude that $K_1\subseteq K_2\subseteq K_3\subseteq\cdots$ and $\cdots I_3\subseteq I_2\subseteq I_1$. Assume that they are all different, then $\dim K_1<\dim K_2<\cdots$. On the other hand, what are the possible values of each? do the same for $I_n$.
(b) First, show that $K_n\subseteq LK_{n+1}$ and $LI_n\subseteq I_{n+1}$. Now, from the choice of $K,I$ it will follow that $LK\subseteq K$ and $LI\subseteq I$. What is $L^N$ restricted to $K$? What is $K\cap L$?
(c) Show that $K\cap L=(0)$. Then use the theorem on dimensions of images and kernels: $\dim\ker T+\dim Im T=\dim V$. Apply it to $L^N$.

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