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 $F[x]$
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?