Let $\alpha$ be an endomorphism of a vector space $V$ of finite dimension $n$.

(a) What is the dimension of the vector space of linear endomorphisms of $V$ ? Show that there exists a non-trivial polynomial $p(X)$ such that $p(\alpha)=0$. Define what is meant by the minimal polynomial $m_{\alpha}$ of $\alpha$.

(b) Show that the eigenvalues of $\alpha$ are precisely the roots of the minimal polynomial of $\alpha$.

(c) Let $W$ be a subspace of $V$ such that $\alpha(W) \subseteq W$ and let $\beta$ be the restriction of $\alpha$ to $W$. Show that $m_{\beta}$ divides $m_{\alpha}$.

(d) Give an example of an endomorphism $\alpha$ and a subspace $W$ as in (c) not equal to $V$ for which $m_{\alpha}=m_{\beta}$, and $\operatorname{deg}\left(m_{\alpha}\right)>1$.