Question

2. Let w-a b :a 2b-7c, a subset of M2 2x2 c d a) (5 points) Find a set of "vectors" in M2-2whose span is W b) (5 points) Show that the vectors in part (a) are linearly independent. c) ( point) Since Wis the span of the vectors obtained in part (a), we can conclude that W isa of M2x2. (See Theorem 4.1.2 in the 4.1 Part 2 notes)

Answer 1

"Given that The Given matrix is w = {[a c b d]: a = 2b - 7c} A subset of M_2 times 2 Find a set of ""vectors"" in M_2 times 2 whose span is w: w = [2b - 7c c b d] = b [2 0 1 0] + c [-7 1 0 0] + d [0 0 0 1] set of vectors in M_2 times 2 = {[2 0 0 1] [-7 1 0 0] [0 0 0 1]} \r\n To show linearly independent we have to show x_1 (2 0 1 0) + x_2 (-7 1 0 0) + x_3 (0 0 0 1) = (0 0 0 0) x_1 = x_2 = x_3 = 0 so now, [2x_1 0 x_1 0] + [-7x_2 x_2 0 0] + [0 0 0 x_3] = [0 0 0 0] [2x - 7x_2 x_2 x_1 x_3] = [0 0 0 0] on comparison we get "