I am studying for a test, and this is one of the practice problems.
Find a basis for the set of vectors in $\mathbb{R}^4$ in the subspace (hyperplane) $x_1 +x_2 + 2x_3 + x_4 = 0, x_1 + 2x_2-x_3=0$
Can I say that the second plane is a linear combination of the first plane, and a basis for the first plane is $\{\begin{bmatrix} 1 & 0 & 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 & 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 2 & -2 \end{bmatrix}\}$, thus it is the basis for the hyperplane (both planes) in the subspace? If not, how do I find the basis?