Linear algebra question?

Let A, B ∈ Mn,n(F) be such that AB = BA and let λ be an eigenvalue of A with eigenvector
~v. Prove the following:
(1) For all m ∈ N and a_0, a_1, . . . , a_m ∈ F, a_mλ^m + a_(m−1)λ^(m−1) + · · · + a_1λ + a_0 is an eigenvalue of a_mA^m + a_(m−1)A^(m−1) + · · · + a_1A + a_0I_n with eigenvector ~v.
(2) B~v ∈ EA(λ).
4 answers 4