Let be an endomorphism of a vector space of finite dimension .
(a) What is the dimension of the vector space of linear endomorphisms of ? Show that there exists a non-trivial polynomial such that . Define what is meant by the minimal polynomial of .
(b) Show that the eigenvalues of are precisely the roots of the minimal polynomial of .
(c) Let be a subspace of such that and let be the restriction of to . Show that divides .
(d) Give an example of an endomorphism and a subspace as in (c) not equal to for which , and .