What's the null space of [111, 000, 000]?

Question

This is the matrix after RREF:

[ 1 1 1
  0 0 0
  0 0 0 ]

I can't find the nulll space of this. let $x_1 = -x_2 -x_3$ . let $x_2 = x_2, x_3 = x_3$.

So I think it's

sp([-1, 1, 0], [-1, 0, 1])

But the solution says it is

sp([1, -1, 0], [1, 0, -1])

How did they get that for the solution? (I'm guessing both are correct? since my solution can form the actual solution, and vise-versa?)

Answer 1

Yes , both are correct . Let $u=(-1,1,0)$ and $v=(-1,0,1)$. Then

$span\{u,v\}= span\{-u,-v\}$.