A linear algebra homework question has me stumped.
Let $S$ be a subspace of $\mathbb{R}^4$ which contains all vectors $[x_1, x_2, x_3, x_4]^T$ for which the following must be true: $x_1 + x_2 + x_3 + x_4 = 0$
Find a basis for the subspace $S^\perp$ where all vectors are orthogonal on $S$.
I'm having trouble understanding how to proceed with this. I first tried to write 4 vectors and use the Gram-Schmidt to construct this basis, but I can't seem to find 4 linear independent vectors. This was the only solution from my perspective but I can't seem to figure it out.
Thanks
Hint We can write the equation $x_1 + x_2 + x_3 + x_4 = 0$ defining $S$ as $$[x_1, x_2, x_3, x_4]^{\top} \cdot [1, 1, 1, 1]^{\top} = 0.$$