Question

1. Let A be an m x n matrix. Determine whether each of the following are TRUE always or FALSE sometimes. If TRUE explain why. If FALSE give an example where it fails. (a) If m n there is at most one solution to Ax = b. always solve Ax b (b) If n > m you can (c) If n > m the null space of A has dimension greater than zero. (d) If n< m then for some b there is no solution of Ax b (e) If n<k the only solution of Ax 0 is x 0

Answer 1

a) False lat A = [1 1 2 2] & b = [2 4] Ax = b has infinity solution. b) No, FALSE A = [1 1 1 2 2 2] & b = [3 4] System has no solution. c) False. A = [1 2] b = [3] Null space has lim1. D) True. As we can have almost m alpha I. rows one of the row must be alpha could be of other. Let v = alpha_1 v_1 + Ellipsis alpha_m v_m = 0 v - alpha_1 v_1 Ellipsis alpha_m v_m = 0 so alpha elementary row operation will make a row zero. choose b s.t corresponding b_i is non -zero e) K is defined (Incorrect questions)

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