Determining if $W$ is a subspace of $\mathbb R^3$ I'm trying to determine whether $W = {\{(x, y, z) | x = 3y, z = -y}\}$ a subspace of $\mathbb R^3$.
If someone can help me understand how to go about doing this that would be great!

As an alternative to performing a subspace test, we can just note that it's a span:  we see that
\begin{align*}
W &= \{(3y,y,-y):y \in \mathbb{R}\} \\
  &=\mathrm{span}\{(3,1,-1)\}
\end{align*}
and spans are subspaces.
