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Linear transformation $T\colon V \to V$ has the property that there is no non-trivial subspace $W$ for which $T(W) \subseteq W$ . Prove that for every polynomial $P$ , $P(T)$ is either invertible or zero.

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    The entire space is itself a non-trivial subspace $W$ for which $T(W) \subseteq W$. I'm certain that non-trivial *proper* subspaces are what was intended.2012-02-16

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Hint: show that $\ker P(T)$ is a linear invariant subspace of $V$ using the fact that $TP(T)=P(T)T$.

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    Well, it will only be proper if n > 1, of course.2012-02-16