Find a basis for a solution set of a linear system

Check your expression for $\vec x$. I think it should be $\vec x=\begin{bmatrix} x_1 \\x_2\\x_3\end{bmatrix}=\begin{bmatrix} -\frac{1}{3}r-\frac{1}{3}s \\r \\s \end{bmatrix}=r\begin{bmatrix} -\frac{1}{3} \\1 \\0 \end{bmatrix}+s\begin{bmatrix} -\frac{1}{3} \\0 \\1 \end{bmatrix}$