Direct product of the kernel and image of a linear transformation

Question

I am practicing QUAL problems for my exam in August and came across this one:

Let $V$ be a finite dimensional vector space over a field and $T:V \rightarrow V$ a linear transformation. Show that there exists $n\ge 1$ such that $V=ker(T^n)\bigoplus im(T^n)$.

First off, I don't understand why this doesn't work for $n=1$? Secondly, where would I even start my thinking in this problem?

Any hints or insight is appreciated! Thanks.

take $V=\langle v_1, v_2\rangle$ and $T(v_1)=v_2, T(v_2)=0$. It doesn't work for $n=1$. Your best bet is to take $n$ large enough so that both of the spaces are stable (and it should work). — Quang Hoang, Jun 05 '15 at 16:08

score 1 · Accepted Answer · answered Jun 06 '15 at 02:21

1

Hint: Consider the Jordan Canonical form of the matrix. See what happens with the block associated with $\lambda = 0$.

answered Jun 06 '15 at 02:21

Ben Grossmann

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Direct product of the kernel and image of a linear transformation

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