Answer
$W$ is not a subspace of $V$.
Work Step by Step
Takinng $a=1$ and $b=0$ we have the vector
$
\begin{bmatrix}
1 \\
0\\
1
\end{bmatrix} \in W.
$
However taking the scalar $c=-1$ in Theorem 6.2 (condition b) we have
$
c \begin{bmatrix}
1 \\
0\\
1
\end{bmatrix} = \begin{bmatrix}
-1 \\
0\\
-1
\end{bmatrix}.$
Now $ \begin{bmatrix}
-1 \\
0\\
-1
\end{bmatrix} \notin W$ as the third coordinate of vectors in $W$ is an absolute value and thus never negative.
