MATH 3319-001 Sample Exam 1a, Fall 2022 Solve the following problems in the space provided. Show your work/reasoning and indicate your nal answer clearly. Indicate problems continued on additional sheets of paper and label those sheets carefully with name and problem number. No calculators, cell phones, or other technology permitted. You may use one 5"x7" card with information written on it. 1. 01 Use Gauss-Jordan elimination to nd the solution set for the system :131 2173 1 3, 33:1 21:2 43:3 = 9, 131 4172 +2333 : 5. . Use Gauss-Jordan elimination to nd the solution set for the system 31'] 2332 = 1, 21m + 1'2 + 5.133 = 4, 73'1 5.132 83:3 = 3. . For A as below: (a) Give values of .'r, y such that :1: 7 y and A2 is the identity matrix. (b) Give values of 1:, y such that .r at y and A2 is the zero matrix. A=liil . (a) Is the set S = {1 + I + 3,2,1 7 3:3, :1: + 3:2 + 2:3, 2:3} C Pg linearly independent? If so, prove it. If not, give a subset of S which is linearly independent and has the same span as S. (b) Does S span Pg? If so, prove it. If not, describe in words span S. . Give an example of matrices A and B (at least 2x2 in size) for which (a) det{2A2) : 12, or else explain why it is impossible; (b) det(B2) = 41 or else explain why it is impossible. . Give a basis for the set of all 3x3 skew-symmetric matrices. . Is W = {(3}, y, z) : a: i y i z = 4} a vector space? If so, give a basis for it. If not, prove not. . True or false? If true, explain why. If false, correct. the statement. The solutions to any linear system of algebraic equations A5 = b form a vector space