Q8. (10 points) Let A be an n x n matrix with eigenvalue and eigenvector Av = Av. We define A0 = I. Suppose that p is a polynomial. Prove that p()) is an eigenvalue of p(A). Q9. Let W be a subspace of V and T : V - V be a linear map. (a) (4 points) Prove that U = {v EV : T(v) E W} is a subspace of V. (b) (3 points) Determine U when W = {0}. (c) (3 points) Determine U when W = V. Q10. (10 points) Suppose that T : V - V is an onto linear map. Show the following: If V = span{v1, . .., Un} then V = span{T(v1), . . . T(Un) }. Page 2