Question

3. You are given the following matrix -4 12 2 7 a)4 points) Find a basis for the nullspace of (b) 4 points] Using the columns of A, find a basis for the column space of A (c) [2 points What are the dimensions of these spaces? (d) [2 points] ls the vector u-I1-1 0 ојт in the nullspace of A? Why? (e) [4 points] Is the vector w-17-9 9-9]T İn the column space of A? If so, express w as a A. linear combination of the basis vectors of the column space of A you found in part (b). () [4 points We want to solve the system Ax b, where b bb b b4T. Find conditions on b, b2, bs, and b to guarantee the existence of a solution to Ax b. Will the solution be unique under these conditions?

Answer 1

Given matrix A = [1 -3 2 -4 -3 9 -1 5 2 -6 4 -3 2 -6 4 -3 -4 12 2 7] The basis for the nullspace of A N(A) = {(x_1, x_2, x_3, x_4) = vector x (-R^4)/vactor Ax = vactor 0} Augment matrix of system vactor Ax = vactor 0 [1 -3 2 -4 0 -3 9 -1 5 0 2 -6 4 -3 0 -4 12 2 7 0] rightarrow Row reduced each form [1 -3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0] Hence x_1 - 3x_2 = 0 x_3 = 0 x_4 = 0, x_4 = 3 x_2 x_3 = 0 = x_4 \r\n N(A) = {[3 lambda, lambda, 0, 0] elements R^n/lambda lelmentof R} = span {(3, 1, 0, 0)} using the columns of A, find a basis for the column space of A by the Evaluation of the precious problem, we see that