How do i find the basis of subspace U

[subopolois]

Homework Statement

In this case find the basis of subspace U
1 2 3 4
5 6 7 8
-6 -8 -10 12

Homework Equations

elementary row operations

The Attempt at a Solution

alright, so i know i have to reduce the matrix and i have done so
1 2 3 4
0 1 2 3
0 0 0 1
now the answer i get is
[1 5 -6]^T [2 6 -8]^T [4 8 12]^T
but the answer in the back of my textbook is
[1 5 -6]^T [2 6 8]^T
what have i done wrong, it seems they have one less column, is it my reduction of the matrix?

 

[stegz]

You just had an error in reducing your matrix somewhere. Check over your work, because the matrix should have been
1 0 -1 -2
0 1 2 3
0 0 0 0
it was probably a simple algebra mistake.

 

[HallsofIvy]

What "subspace U" are you talking about? You give an array of numbers which you refer to as a matrix. I can think of 4 different subspaces that might be meant here:

The row space of the matrix.
The column space of the matrix.
The null space (kernel) of the matrix.
The image of the matrix.

Since you are row-reducing the matrix, you probably mean the column space but it would be a good idea to say so!