Q:

Finding a basis for a given subspace of $\Bbb R^4$

Find a basis for the subspace $ W = \{(x, y, z, w) \in\Bbb R^4
: y − 2z + w = 0\}$. What is $\dim(W)$?
I don't seem to understand how to solve this problem. I just don't know where to start I am not given an A matrix to work with. possibly the b matrix is $[0, 1, -2,1]$ but is that enough to find the basis? 

A:

Well, every vector in $W$ may be written as
$$
\begin{bmatrix}
x\\ y\\ z\\ 2z-y
\end{bmatrix}
=x
\begin{bmatrix}
1\\0\\0\\0
\end{bmatrix}
+
y
\begin{bmatrix}
0\\ 1\\ 0\\ -1
\end{bmatrix}
+
z
\begin{bmatrix}
0\\0\\1\\ 2
\end{bmatrix}
$$
One easily checks that the collection
$$
\mathcal B=
\left\{
\begin{bmatrix}
1\\0\\0\\0
\end{bmatrix}
,
\begin{bmatrix}
0\\ 1\\ 0\\ -1
\end{bmatrix}
,
\begin{bmatrix}
0\\0\\1\\ 2
\end{bmatrix}
\right\}
$$
is linearly independent. Hence $\mathcal B$ is a basis for $W$ which implies $\dim(W)=3$.

