Question

Find bases for the four fundamental subspaces of the matrix A 1 4 9 0 20 N(A)-basis NCAT) = R(A)-basis R (A' )-basis

Answer 1

Given that A = [1 0 4 2 9 0] it's row reduce (A) = [1 0 4 2 9 0] - (1) N(A) - basis: for find basis for null space we see that the general solution AX = 0 is let X = [x_1 x_2 x_3] Rightarrow [1 0 4 2 9 0] [x_1 x_2 x_3] = [0 0 0] x_1 + 4 x_2 + 9 x_3 = 0 2 x_2 = 0 Rightarrow x_2 = 0 Rightarrow x_1 = -9 x_3 x_2 = 0 x_3 = x_3] Here only x_2 is free variable X = [x_1 x_2 x_3] = [-9 x_3 0 x_3] = x_3 [-1 0 1] Forall x elementof R^2 = span {[-9 0 1]} Thus the bet is N(A) - Basis = span [-9 0 1] is only 1.5 meter so N(A) Basis = [-9 0 1] R(A) - Basis: non zero rows of row