Find an orthogonal basis for $\mathbb R^4$ Find orthogonal basis for $\mathbb R^4$ that contains the vectors: $v_1=$
$\begin{bmatrix}
       2            \\[0.3em]
       1            \\[0.3em]
       0            \\[0.3em]
       -1
     \end{bmatrix}$ and $v_2=$
$\begin{bmatrix}
       1            \\[0.3em]
       0            \\[0.3em]
       3            \\[0.3em]
       2
     \end{bmatrix}$
Right now I'm trying to find any basis for $\mathbb R^4$ that contains $v_1$ and $v_2$. Multiplying both equations by $\begin{bmatrix}
       x            \\[0.3em]
       y            \\[0.3em]
       z            \\[0.3em]
       w
     \end{bmatrix}$ I got $2x + y -z = 0 | x + 3y + 2z = 0$. What to do from here? How can I find these vectors so I can proceed to applying the Gram-Schmidt Process?

By considering linear combinations we see that the second and third entries of $v_1$ and $v_2$ are linearly independent, so we just need $e_1=(1,0,0,0)^T,e_4=(0,0,0,1)$ To form an orthogonal basis, they need all be unit vectors, as you are mot asked to find an orthonormal basi.
