Question

6. Construct a matrix whose column space contains (1, 1,5) and (0,3,1) and whose nullspace contains

Answer 1

Since (1,1,5) and (0,3,1) are in column space we can take them as the first column and the second column of the required matrix Let the third column be (y_1 y_2 y_3) so the matrix looks like (1 1 5 0 3 1 y_1 y_2 y_3) Since (1 1 2) is the null space of the matrix we have (1 1 5 0 3 1 y_1 y_2 y_3)(1 1 0) = (0 0 0) = > 1 + 2y_1 = 0, n + 2y_2 = 0, 6 + 2y_2 = 0 = > y_1 = -1/2 y_2 = -2 y_3 = -3 So the required matrix is (1 1 5 0 3 1 -1/2 -2 -3)