Answer
W is not a vector space.
Work Step by Step
$W\subset \mathbb{R}^{3}$.
In order to be a subspace of $\mathbb{R}^{3}$ (definition, p.195) , W must,
(a) contain zero,
(b) be closed over addition
(c) be closed over multiplication by scalars.
Checking,
(a) fails, because $5(0)-1\neq 0+2(0),$
$\left[\begin{array}{l}
0\\
0\\
0
\end{array}\right]\not\in W$
so, W is not a subspace of $\mathbb{R}^{3}$,
W is not a vector space.