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.
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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4 years ago