# minimal polynomials for linear algebra

Q:

let $T$ be a linear transformation on V over field $F$. Let c be a characteristic value of T and $W_c$ the characteristic subspace of T associated with c. Suppose a proper T-invariant subspace $W \supset W_c$, and there is a vector $\alpha$ in V not in W, such that $(T-cI)\alpha \in W$. Prove that the minimal polynomial $p_T$ of T is in the form of $(x-c)^2$q for nonzero polynomial q in $F[x]$

Progress:

My first thought is to use $p_\alpha$|$p_T$, and I need to show $(T-cI)^2\alpha=0$. for a certain vector $\alpha$ in V\W, $(T-cI)\alpha$ is not zero because otherwise $\alpha$ is in $W_c$ and thus is in W (contradict).
So if I can show $W=W_c$, then it's done. But I fail to show W=Wc. Or am I in the wrong way?

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Out of curiosity, why delete and repost? –  Jonas Meyer Aug 1 '12 at 1:50
I corrected some writting, it didnot get answer for the day –  David Aug 1 '12 at 3:30
To correct writing, editing seems to work well. There is no need to delete and repost. This thread on meta might be relevant. –  Jonas Meyer Aug 1 '12 at 3:34
What is $p_\alpha$? $W$ does not have to equal $W_c$. –  Jonas Meyer Aug 1 '12 at 3:39
$p_\alpha$ is the T-annihilator associated to $\alpha$, which divides $p_T$. can you offer me some hints? –  David Aug 1 '12 at 17:02
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