Question

Problem 2. Find a vector 7 orthogonal to the row space, and a vector y orthogonal to the column space of the matrix [1 2 1] 2 4 3 [36 4

Answer 1

2. Let the given matrix be denoted by A. The RREF of A is

1	2	0
0	0	1
0	0	0

This implies that:

i). The set {(1,2,0),(0,0,1)} is a basis for the row space of A.

ii).

The 1^st and the 3^rd columns of A are linearly independent and the 2^nd column is a scalar multiple of the 1^st column.

Hence the set{(1,2,3)^T,(1,3,4)^T} is a basis for the column space of A.

Now, let X = (x,y,z) be an arbitrary vector orthogonal to the row space of A. Then (x,y,z).(1,2,0) = 0 or, x+2y = 0 or, x = -2y and (x,y,z).( 0,0,1) = 0 or, z = 0. Then X = (-2y,y,0) = y(-2,1,0). Assuming y = 1, X = (-2,1,0) is a vector orthogonal to the row space of A.

Further, let Y = (x,y,z)^T be an arbitrary vector orthogonal to the column space of A. Then (x,y,z)^T.( 1,2,3)^T = 0 or, x+2y+3z = 0 and (x,y,z)^T.( 1,3,4)^T = 0 or, x+3y+4z = 0.

On solving these equations, we get x = -z and y = -z so that Y = (-z,-z,z)^T = z(-1,-1,1)^T. Assuming z = 1, Y = (-1,-1,1)^T is a vector orthogonal to the column space of A.