# Property of an operator in a finite-dimensional vector space $V$ over $R$

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) Prove that 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. 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) Prove that $V = K \oplus I$.

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

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Just view matrices as $m$-tuples of column vectors in $L$. – Hagen von Eitzen Dec 13 '12 at 22:40
??????????????????????????????????????? – Mike Dec 14 '12 at 3:47
What have you done? Where are you stuck? – Olivier Bégassat Dec 14 '12 at 4:07
Try thinking about how the $R$-action acts on $L$ and $R$.
Look at the $R$-homomorphism sending $(l_1,...,l_n)\in L^n$ to $(l_1|...|l_n)=A\in R$, such that $l_i$ is the $i$-th column vector of $A$.
Do you see how this map gives an isomorphism of $R$-modules?