Find an orthonormal basis which includes the vectors $u_1 = (\frac{1}{\sqrt2}, 0,\frac{1}{\sqrt2}, 0)$ and $u_2 $

Question

I have to find an orthonormal basis for $ \Bbb R^4 $ which includes the vectors $u_1 = (\frac{1}{\sqrt2}, 0,\frac{1}{\sqrt2}, 0)$ and $u_2 = (\frac{-1}2, \frac1 2,\frac1 2, \frac{-1}2)$

I really do not know how to do it, I tried to use the Gram-Schmidt method but when I started with $u_1$ then obtained $u_2$.

Any ideas?

Answer 1

Aside from the normalization constants, the vectors can be $\left(1,0,1,0\right)$, $\left(-1,1,1,-1\right)$, $\left(0,1,0,1\right)$, $\left(-1,-1,1,1\right)$