How to find a set of vectors spanning the solution space of $Ax=0$, where How to find a set of vectors spanning the solution space of $Ax=0$, where

Basically I have tried many times to solve it and my answer consistently comes in the following form:
$\pmatrix{1 \\ -1 \\-1\\0}$
While my book gives an answer of:
$\pmatrix{-1 \\-1\\1\\0}$

Do the following row-operations: 
$$R_2:-R_4, \; R_3:-2R_1, \; R_4: -R_1, \; R_4:-R_3, \; R_3: -R_2$$
which gives you 
$$\pmatrix{1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0}$$
which means $x_1+x_3=0, x_2+x_3=0, x_4=0$. Letting $x_1=x_2=t$, we have $$x=\pmatrix{t \\ t \\ -t \\ 0} \rightarrow \pmatrix{1 \\ 1 \\ -1 \\ 0}$$
