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If $T$ is an endomorphism of a finite-dimensional vector space $V$ over a finite field, then how can I show that there exists a positive integer $r$ such that $T^r$ is a projection operator?

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    Please do not ask questions in the imperative, you make it sound like a homework question. And if it is indeed a homework question, please first read our FAQ on that: http://meta.math.stackexchange.com/q/1803/15432011-08-13
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    Do you know what is a Jordan normal form?2011-08-13
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    @Sarah I tried to fix up the question a little bit (the problems were not mathematical). Hope that's alright!2011-08-13
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    @Prometheus, if I remember correctly, you can't use the Jordan normal form for matrices over finite fields.2011-08-13
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    @Sarah: what have you tried so far? What tools did you consider?2011-08-13
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    @Calle Are you certain? The proofs I've seen of the JNF only rely on the structure theorem for modules over a PID, and $k[x]$ is a PID for any field $k$. You need the characteristic polynomial to split, but this only requires moving up to a larger finite field.2011-08-13

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