Name : ID# Section Math 227 Spring 2024 Homework 4 due May 19 ● Submit the homework to the course moodle page no later than 11:55 pm May 19. Late homework will not be accepted. There is no make up for homework. • Show all your work and explain your reasoning in full. Write your solutions in well-organized English and mathematical sentences. • Similarities between papers of different students that cannot be explained as coincidence will be treated as cheating. Please do not write below this line. Q1 Q2 Q3 Q4 Q5 TOTAL 20 20 20 20 20 20 100 1) Find bases for the row and column and null spaces of the matrix -2 -2 -3 2 2 4-2 342 1 0 4 1 -5 -7 What is the rank and nullity of the matrix? 1 2. a) Let A be a 4 × 6 matrix. What is the largest possible value for rank of A? Explain. Prove or disprove that the columns of A are always linearly dependent. b) Let Rn denote the space of row matrices with n entries. Suppose that L : R4 → R3 is the linear transformation defined by L([a, b, c, d]) = [a+b, c−d, a+3c, a+2b+d]. Find the 3 × 4-matrix A associated to L with respect to the standard basis for R and R3. 3) a) Prove that if A is invertible and diagonalizable, then A¯¹ is also diagonalizable. b) Let S and T be two bases for the finite dimensional vector space V. Show that the transition matrix Prs is invertible. c) True or false? Let A be an n × n invertible matrix. Then A is the transition matrix Pr←s with respect to some bases S and T of Rn. Explain. 4) Let -5 A = 0 1 0 6 -10 If possible, find an invertible matrix P such that P-¹AP is diagonal.