Answer
$S$ is a subspace.
Work Step by Step
If $y=0, z=2x$. then $\left[\begin{array}{r}{x} \\ {y}\\{z}\end{array}\right]=x\left[\begin{array}{r}{1} \\ {0}\\{2}\end{array}\right]$ and hence $S$ is spanned by the vector $\left[\begin{array}{r}{1}\\ {0}\\{2}\end{array}\right]$ by Theorem 3.19 $S$ is a subspace.
