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:
$\pmatrix{-1 \\-1\\1\\0}$

Reducing:
$$\begin{pmatrix}1&0&1&0\\1&2&3&1\\2&1&3&1\\1&1&2&1\end{pmatrix}\longrightarrow\begin{pmatrix}1&0&1&0\\0&2&2&1\\0&1&1&1\\0&1&1&1\end{pmatrix}\longrightarrow\begin{pmatrix}1&0&1&0\\0&1&1&1\\0&0&0&\!\!-1\\0&0&0&0\end{pmatrix}$$
The general solution to the homogeneous system $\;A\vec x=\vec0\;$ is
$$x_4=0\;,\;\;x_1=x_2=-x_3\;\implies\;\left\{\;\begin{pmatrix}t\\t\\\!\!-t\\0\end{pmatrix}\;\;;\;\;\;t\in\Bbb F\right\}$$
whatever the field $\;\Bbb F\;$ is. Thus, a particular non-zero solution, and also a basis for the solution space, is for example
$$\begin{pmatrix}1\\1\\\!\!-1\\0\end{pmatrix}$$
