Null space of a matrix

Question

A is a matrix which has two special solutions to Ax=0, all other solutions are linear combinations of the special solutions. They are (3,1,4,0,5) and (2,0,2,1,2).

C is a matrix that is the same as A except its second column is (col 2 of A) - (col 1 of A). What is a basis for the nullspace of C?

Answer 1

C = A • T

where T = Matrix {{1, -1, 0, 0, 0}, {0,1, 0, 0, 0}, {0, 0,1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0,1}}

Given X as solution for A X = 0 ,the Solution for C X' = 0 will be X' = T^-1 X since

C X' = C (T^-1X) = A•T (T^-1X) = A X = 0

T^-1 = Matrix {{1, 1, 0, 0, 0}, {0,1, 0, 0, 0}, {0, 0,1, 0, 0},{0, 0, 0, 1, 0}, {0, 0, 0, 0,1}}.

T is a 5x5 matrix. Only the top left 2x2 sub matrix of T is non-diagonal and its inverse can be quickly found using the 2x2 matrix inversion. The remaining lower right 3x3 sub portion of T are identity matrix and the inverse is also identity matrix.

X' solution can be obtained

T^-1 (3,1,4,0,5) = (4,1,4,0,5)

T^-1 (2,0,2,1,2) = (2,0,2,1,2)

New basis are (4,1,4,0,5) and (2,0,2,1,2)