Finding orthogonal basis in $\mathbb R^4$ from given vectors

Question

I have two subsets of $\mathbb R^4$

$S=((-1,0,1,1),(0,1,1,1),(1,0,0,1))$

and $T=(x,y,z,x-y+2z)$

I've proved that T is a subspace of $\mathbb R^4$ and that S is a basis for T. So far, so good!

I have to show that 2 of the vectors are orthogonal, which I have done, namely $(-1,0,1,1)$ and $(1,0,0,1)$ as their product is equal to zero. I now need to find an orthogonal basis for "T".

I presume I need to use Gram-Schmidt process, but am struggling as to how to find two more orthogonal vectors to start with. Would any starting vector need to be in the form $(x,y,z,x-y+2z)$?

Any hints/help much appreciated!

Answer 1

Why two more vectors? Since $\dim T=3$, you only need one more vector. So, apply Gram-Schmidt to $e_1=(-1,0,1,1)$, $e_2=(1,0,0,1)$, and $e_3=(0,1,1,1)$. The fact that $e_1$ and $e_2$ are orthogonal will make your computations easier.

Answer 2

If "S is basis for T" and S has only 3 vektor in it, then you can assume, that dim(T) = 3 so you don't need two more vektors. Just use the Gram-Schmidt process on those 3 basevektors in S. But as we know that two of them are orthogonal, you just need to sove the last one.