How to determine if set of vectors is a basis for W Consider the subspace $$
W =\left\{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3}\end{bmatrix} \in \mathbb{R}^3 \,| x_{1}+x_{2}+x_{3} = 0 \right\} 
$$
Is the set S a basis for W? $$
S= \left\{ \begin{bmatrix} -1 \\ -1 \\ 2 \end{bmatrix} , \begin{bmatrix} -3 \\ 2 \\ 1\end{bmatrix} \right\}
$$
I'm not too sure how to go about this question but this is what I've tried. Let $x_{2} = s$ and $x_{3}=t$. Then I have $$
\begin{bmatrix} -s-t \\ s \\ t\end{bmatrix} =s \begin{bmatrix} -1 \\ 1 \\ 0\end{bmatrix} + t \begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}
$$
Therefore, the basis for $W$ is $$
A= \begin{bmatrix} -1 \\ 1 \\ 0\end{bmatrix} , \begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}
$$
Since the vectors in $S$ is not a scalar multiple of the vectors in $A$, $S$ is not a basis for $W$.
Could anyone confirm if I am correct? Thanks

The vectors are from $W$ and they are linearly independent, hence they are basis of $W$, as $\dim W=2$.
