Solution
= Solution
Take $V=\mathbb R^3$, $\alpha=\operatorname{diag}(0,1,0)$ and $W=\operatorname{span}(e_1,e_2)$. Then $W$ is a proper <invariant subspace>, and the restriction has matrix $\operatorname{diag}(0,1)$. In each case $X(X-1)$ annihilates the operator, while any annihilator must vanish at both zero and one. Consequently
$$
\boxed{m_\alpha(X)=m_\beta(X)=X(X-1),\qquad\deg m_\alpha=2>1}.
$$
This same example works over any scalar field.