Answer
The set $W$ is a subspace of $R^{n}$.
Work Step by Step
Let $A$ be a fixed $m\times n$ matrix and $x, y\in W$, then $Ax=0$ and $Ay=0$
Thus we have that
1) $A(x+y)=Ax+Ay=0+0=0 $
this shows that $x+y\in W$
2) $A(\alpha x)=\alpha (Ax)=\alpha . 0=0$ $\implies$ $ \alpha x\in W$
3) There is the zero vector $0\in W$ such that $A. 0=0$
4) For all vector $x\in W$ , there is an $(-x) \in W$ such that
$A(x+(-x))=Ax +A(-x)=Ax-Ax=0$
From $1),$ $ 2),$ $3),$ $ 4),$ , we see that
the set $W$ is a subspace of $R^{n}$.
