Answer
$ w\in$Col A
and
$ w\in$Nul A.
Work Step by Step
A typical vector v in Nul A has the property that Av=0.
A typical vector v in Col A has the property that the equation Ax=v is consistent.
Is $Ax=w$ consistent$?$
[A w]=$\left[\begin{array}{lll}
-6 & 12 & 2\\
-3 & 6 & 1
\end{array}\right]\left[\begin{array}{l}
.\\
-2R_{1}
\end{array}\right]$
$\sim\left[\begin{array}{lll}
-6 & 12 & 2\\
0 & 0 & 0
\end{array}\right]$...
The system is consistent, so $ w\in$Col A.
Is Aw=0?
$\left[\begin{array}{ll}
-6 & 12\\
-3 & 6
\end{array}\right]\left[\begin{array}{l}
2\\
1
\end{array}\right]$=$\left[\begin{array}{l}
-12+12\\
-6+6
\end{array}\right]=\left[\begin{array}{l}
0\\
0
\end{array}\right]$
Yes. $ w\in$Nul A
