MATH 2350 Assignment 5 Due Friday April 7th at 16:00 7 days Problem 1: Let T : P2 (R) P2 (R) be defined by T (P ) = P (X + 2) . Find the matrix of T with respect to the canonical basis 1, X, X 2 . Problem 2: On P2 we use the inner product < P, Q >= P (0)Q(0) + P (1)Q(1) + P (3)Q(3) . Apply Gram Schmidt to 1, X, X 2 . Problem 3: On M3,2 consider the inner product < A, B >= tr(AT B) Calculate and 2 < 1 2 1 3 2 , 2 1 1 1 2 > 2 1 1 k 1 1 k 2 2 Problem 4: Find the least square solution for Ax = b where 4 1 2 1 0 3 A= 2 5 ; b = 2 4 3 0 Problem 5: Find the line which best fits the points (5, 3); (0, 3); (5, 2); (10, 0). Calculate the corresponding least square error. Problem 6: Find the singular values 0 A = 1 2 Problem 7: Find the least square 2 0 1 2 A= 2 1 0 1 of 1 0 2 solution for Ax = b where 1 0 6 2 ;b= 0 0 1 6 1 Problem 8: If (V, < , >) is an inner product space and u, v V , prove that ku + vk2 + ku vk2 = 2(kuk2 + kvk2 ) . 2