Answer
$\left[\begin{array}{l}
0\\
0\\
0
\end{array}\right]\not\in W$
$W$ is not a subspace of $\mathbb{R}^{3}.$
Work Step by Step
See the definition of a subspace on page 195.
(a) Is the zero vector of $\mathbb{R}^{3}$ in $W$?
Do a,b$\in \mathbb{R}$ exist such that
$\left[\begin{array}{l}
0\\
0\\
0
\end{array}\right]=\left[\begin{array}{l}
3a+b\\
4\\
a-5b
\end{array}\right]$?
This vector equality implies a linear system of three equations,
of which the second equation is
$0=4$,
making the system inconsistent (has no solutions)
so, $\left[\begin{array}{l}
0\\
0\\
0
\end{array}\right]\not\in W$
$W$ is not a subspace of $\mathbb{R}^{3}.$
