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
$\left\{\begin{array}{l}
x_{1}-2x_{2}-4x_{3}=0\\
2x_{1}-x_{3}-3x_{4}=0
\end{array}\right.$
That is,
W=Nul ($\left[\begin{array}{llll}
1 & -2 & 3 & 0\\
2 & 0 & -1 & -3
\end{array}\right]$)
By Theorem2,
The null space of an m$\times$n matrix A is a subspace of $\mathbb{R}^{n}$.
So,
W is a vector space.