Answer
By Theorem 3, W is a vector space.
Work Step by Step
$W\subset \mathbb{R}^{3}$.
$u\in W\Rightarrow u=a\left[\begin{array}{l}
-1\\
1\\
3
\end{array}\right]+b\left[\begin{array}{l}
2\\
-2\\
-6
\end{array}\right]$
$W=$Span $\{ \left[\begin{array}{l}
-1\\
1\\
3
\end{array}\right],\ \left[\begin{array}{l}
-1\\
1\\
3
\end{array}\right] \}$
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=$\left[\begin{array}{ll}
-1 & 2\\
1 & 1\\
3 & 3
\end{array}\right]$
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.