Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

Hints:
(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 $L^N$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.