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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.

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  • $\begingroup$ 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). $\endgroup$ Commented Jun 5, 2015 at 16:08
  • $\begingroup$ this might help! $\endgroup$ Commented Jun 5, 2015 at 16:28

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

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