Answer
$u=\left[\begin{array}{l}
1\\
0\\
-1\\
0
\end{array}\right], v=\left[\begin{array}{l}
-1\\
1\\
0\\
1
\end{array}\right],w=\left[\begin{array}{l}
0\\
-1\\
1\\
0
\end{array}\right].$
$W=$Span$ \{u,v,w\}$,
Work Step by Step
See Theorem 1, p.196
If $v_{1},v_{2}...,v_{p}$ are in a vector space $V$, then
Span $\{v_{1},v_{2}...,v_{p}\}$ is a subspace of V.
------------
$\left[\begin{array}{l}
a-b\\
b-c\\
c-a\\
b
\end{array}\right]=a\left[\begin{array}{l}
1\\
0\\
-1\\
0
\end{array}\right]+b\left[\begin{array}{l}
-1\\
1\\
0\\
1
\end{array}\right]+c\left[\begin{array}{l}
0\\
-1\\
1\\
0
\end{array}\right]$
$u=\left[\begin{array}{l}
1\\
0\\
-1\\
0
\end{array}\right], v=\left[\begin{array}{l}
-1\\
1\\
0\\
1
\end{array}\right],w=\left[\begin{array}{l}
0\\
-1\\
1\\
0
\end{array}\right].$
$u,v,w\in \mathbb{R}^{3}$,and $W=$Span$ \{u,v,w\}$,
so , by Theorem 1,
$W$ is a subspace of $\mathbb{R}^{3}.$
