Answer
By Theorem 3, W is a vector space.
Work Step by Step
$W\subset \mathbb{R}^{3}$.
$u\in W\Rightarrow u=c\left[\begin{array}{l}
1\\
0\\
1
\end{array}\right]+d\left[\begin{array}{l}
-6\\
1\\
0
\end{array}\right]$
$W=$Span $\{ \left[\begin{array}{l}
1\\
0\\
1
\end{array}\right]$, $\left[\begin{array}{l}
-6\\
1\\
0
\end{array}\right] \}$
By definition, on p.203
The column space of an m$\times$n matrix A, written as Col A,
is the set of all linear combinations of the columns of A.
$W=$Col A, where A=$\left[\begin{array}{ll}
1 & -6\\
0 & 1\\
1 & 0
\end{array}\right]$
By Theorem 3,
The column space of an m$\times$n matrix A is a subspace of $\mathbb{R}^{m}$.
So W is a vector space.