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
(a) Is the zero vector of $\mathbb{R}^{3}$ in $W$?
Do a,b$\in \mathbb{R}$ exist such that
$\left[\begin{array}{l}
-a+1\\
a-6b\\
a+2b
\end{array}\right]=\left[\begin{array}{l}
0\\
0\\
0
\end{array}\right]$
This vector equality implies a linear system of three equations,
with the augmented matrix
$\left[\begin{array}{llll}
-1 & 0 & | & -1\\
1 & -6 & | & 0\\
1 & 2 & | & 0
\end{array}\right]\left\{\begin{array}{l}
\times(-1).\\
\leftarrow(R_{2}+R_{1}).\\
\leftarrow(R_{3}+R_{1}).
\end{array}\right.$
$\left[\begin{array}{llll}
1 & 0 & | & 1\\
0 & -6 & | & -1\\
0 & 2 & | & -1
\end{array}\right]\left\{\begin{array}{l}
.\\
\leftarrow(R_{2}+3R_{3}).\\
.
\end{array}\right.$
$\left[\begin{array}{llll}
1 & 0 & | & 1\\
0 & 0 & | & -4\\
0 & 2 & | & -1
\end{array}\right]$
Swapping $R_{2}$ and $R_{3}$ we have
0 0 $|$ -4
in the last row, so this is
an incosistent system.
$\left[\begin{array}{l}
0\\
0\\
0
\end{array}\right]\not\in W$
$W$ is not a subspace of $\mathbb{R}^{3}.$
