Answer
$W$ is not a Subspace of $V$.
Work Step by Step
Taking $a=b=0$, we have that
$\begin{bmatrix}
0\\
0\\
1
\end{bmatrix} \in W.$
However, taking the scalar $c=0$ in Theorem 6.2 (condition b) we have that
$
c
\begin{bmatrix}
0\\
0\\
1
\end{bmatrix}
=
\begin{bmatrix}
0\\
0\\
0
\end{bmatrix} \notin W.
$
