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Let $T:V \rightarrow V$ linear transformation over field $F$. Let $W$ be a non zero proper $T$-invariant subspace of $V$. Suppose that the characteristic polynomial $f_T$ of $T$ satisfies that $f_T(0)\neq 0$. Show that if $T$ has a cyclic vector, then the restricted $T|_W:W \rightarrow W $ has a cyclic vector.

I think I should use that $p_T=f_T$ but how can I connect this with the invariant space?

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    The condition on the characteristic polynomial is redundant; it follows automatically from the fact that $T$ has a cyclic vector.2012-07-31

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