Math 110, Fall 2015. Homework 13, due Nov 25. Prob 1. Let V be a complex n-dimensional space and let T L(V ) be such that null T n3 = null T n2 . How many distinct eigenvalues can T have? Prob 2. Let V = P3 (C) and let D L(V ) be the dierentiation operator. Find a square root of II + D. Prob 3. Let V be a complex (nite-dimensional) vector space and let T L(V ). Prove that there exist operators D and N in L(V ) such that D is diagonalizable, N is nilpotent, and DN = N D. Prob 4. Suppose that V is a complex vector space of dimension n. Let T L(V ) be invertible. Let p denote the characteristic polynomial of T and let q denote the characteristic polynomial of T 1 . Prove that q(z) = 1 zn p p(0) z for all z C. Prob 5. Suppose the Jordan form of an operator T L(V ) consists of Jordan blocks of sizes 3 3, 4 4, 1 1, 5 5, 2 2, corresponding to eigenvalues 1 , 2 , 3 , 2 , 1 , respectively. Assuming that i = j for i = j, nd the minimal and the characteristic polynomial of T