Question

Problem #10: Let 1 1 4 4] 1 2 5 3 -1 0 6 3 1 6 3 (a) Find the dimension of the row space of A. (b) Find the dimension of the nullspace of A. Problem #10(a): Problem #10(b):

Answer 1

We have given A = [1 1 3 3 1 - 1 - 1 1 4 - 2 0 6 - 4 5 6 - 3] we have to find dimension of range space. operating R_2 rightarrow R_2 - R_1, R_3 rightarrow 3 R_1 - R_3, R_4 rightarrow 3 R_1 - R_4 we get [1 0 0 0 1 - 2 - 4 - 2 4 - 6 - 12 - 6 - 4 9 18 9] operating R_3 rightarrow R_3 - 2 R_1, R_4 rightarrow R_4 - R_2, we get [(1) 0 0 0 1 (- 2) 0 0 4 - 6 0 0 - 4 9 0 0] we got only two point in Row echolon form of A. so corresponding two columns are linear Independent and the two rows are L.I \r\n so Basis of Row space has have these two rows i.e Basis for row space = {(1,1,4, - 4), (0, - 2, - 6, 9)} = > dimension for row space is 2. Now Let Ax = 0 [1 1 3 3 1 - 1 - 1 1 4 - 2 0 6 - 4 5 6 3]