Answer
$S$ is a subspace.
Work Step by Step
If $x=0$. then $\left[\begin{array}{r}{x} \\ {y}\end{array}\right]=\left[\begin{array}{r}{0} \\ {y}\end{array}\right]=y\left[\begin{array}{r}{0}\\ {1}\end{array}\right]$ and hence $S$ is spanned by the vector $\left[\begin{array}{r}{0}\\ {1}\end{array}\right]$ by Theorem 3.19
$S$ is a subspace.
