Find an orthonormal basis of the subspace:
$$V = {[x, y, z, w]^T:x+y+z+w=0}$$ of $\mathbb{R}^4$
First I found a $4\times4$ determinant to verify whether they are non-singular or not.
$$\begin{vmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{vmatrix}=1\ne0$$
Then to orthogonalize the basis I applied Gram-Schmidt process,
$$v_1=x_1=(1,1,1,1)$$
Is my basis $(1,1,1,1)$ correct?
Best Answer
Your proposed basis vector is not in your subspace $$x+y+z+w=0$$ because $$1+1+1+1=4 \ne 0$$
you need to find some linearly independent vectors in the subspace to form a basis and then apply Gram-Schmidt method to find an orthonormal basis.
For example $$(1,-1,0,0), (0,1,-1,0), (0,0,1,-1)$$ are linearly independent vectors in your subspace.
Can you apply Gram-Schmidt to that set to find an orthonormal basis?