1. Suppose that T : V - V has dim V distinct eigenvalues and suppose that S : V - V has the same eigenvectors as T (not necessarily with the same eigen values). Prove that ST = TS. 2. Suppose that T : V - V is a linear map. Show that (a) 0 is an eigenvalue for T if and only if T is not invertible (b) If A is an eigenvalue for T, then A" is an eigenvalue for T (c) If T is invertible and A is an eigenvalue for T, then A-is an eigenvalue for T-. Note: V is not necessarily finite dimensional so you cannot take determinants. Use the definition that A is an eigenvalue if and only if T - X/ is not injective. 3. Consider T : R[x]s2 - R[x]s2 defined by (Tf)(x) = (x - y)?f(y)dy - 2f(0)12 for f E R[x]s2. Find all eigenvalues and eigenvectors for T