# finding an orthonormal basis of a subspace

Let $$W = \{(x, y, z, w) \in C^4| x + y − z − w = 0\}$$.

I have proved that this is a subspace (ie, nonempty, closed under scalar multiplication and vector addition).

I have not been able to find any information on how to form an orthonormal basis for a subspace. What I have been doing is Gram-Schmidt for a set of given vectors. My intuition is to switch this to coordinates and form a matrix to perform Gram Schmidt. But, would the coordinates give $$\{(1,0,0,0),(0,1,0,0),(0,0,-1,0),(0,0,0,-1)\}$$? If so, the inner product of this space would be $$0$$ and it is normalized with magnitude$$=1$$. Is there more to this?

Thank you!

The four vectors you listed are a basis of the full $$\Bbb C^4$$; whatever you did with the algebra cannot possibly be right. Think that your space $$W$$ is described by $$1$$ equation, so its dimension is $$4-1=3$$. Now, the equation $$x+y-z-w=0$$ means that $$w = -x-y+z$$, and thus $$\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} = \begin{bmatrix} x \\ y \\ z \\ -x-y+z \end{bmatrix} = x \begin{bmatrix} 1 \\ 0 \\ 0 \\ -1 \end{bmatrix} + y \begin{bmatrix} 0 \\ 1 \\ 0 \\ -1 \end{bmatrix} + z \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}.$$Hence the vectors $$v_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ -1 \end{bmatrix},\qquad v_2= \begin{bmatrix} 0 \\ 1 \\ 0 \\ -1 \end{bmatrix},\qquad\mbox{and}\quad v_3= \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}$$are a basis for $$W$$. Apply Gram-Schmidt to $$\{v_1,v_2,v_3\}$$ instead.
• Is the vector $(2,1,1,1)$ in $W$? Commented Dec 7, 2022 at 2:52