Answer
By Theorem 2, W is a vector space.
Work Step by Step
$W$ is the set of all solutions to
the homogeneous system of equations
$\left\{\begin{array}{l}
x_{1}+3x_{2}-x_{3}=0\\
x_{1}+x_{2}+x_{3}=0
\end{array}\right.$
That is,
W=Nul ($\left[\begin{array}{llll}
1 & 3 & -1 & 0\\
1 & 1 & 1 & 0
\end{array}\right]$)
By Theorem 2,
The null space of an m$\times$n matrix A is a subspace of $\mathbb{R}^{n}$.
So,
W is a vector space.