Math 110, Fall 2015. Homework 6, due Oct 7. Prob 1. Let V = P2 (IR) and suppose j (p) = p(j), j = 0, 1, 2. Prove that (0 , 1 , 2 ) is a basis for P2 (IR) and nd a basis (p0 , p1 , p2 ) of P2 (IR) whose dual is (0 , 1 , 2 ). Prob 2. Let V be a nite-dimensional vector space and let U be its proper subspace (i.e., U = V ). Prove that there exists V such that (u) = 0 for all u U but = 0. Prob 3. Let T : P(IR) P(IR) : p(x) (x 1)3 p(x) + p (x). (a) Let P(IR) : (p) = p (1). Give a formula for T (). (b) Let P(IR) : (p) = 1 0 p(x) dx. Evaluate T ()(x2 ). Prob 4. Suppose V is nite-dimensional and U , W are its subpaces. Prove that (U W )0 = U 0 + W 0 . Prob 5. Suppose V and W are nite-dimensional, T L(V, W ), and null T = span() for some W . Prove that range T = null