Find orthogonal basis of a space

Question

I am trying to find an orthogonal basis of a space W defined by vectors. $ W=[(0,1,0,1),(1,1,0,1),(0,0,0,1)] $ . How would I achieve so? I have no idea how to begin. In my textbook there is a hing to find such vectors, so that they are mutually perpendicular, but that doesn't help me much. Thanks!

Answer 1

Let's see. Gram-schmidt may not be necessary.

The space is at most $3$ dimensional. And one easily gets $(1,0,0,0)$ and $(0,1,0,0)$ in addition to $(0,0,0,1)$ in the span.

Answer 2

If you apply the Gramm-Schmidt process to your set of vectors, then you will get$$\left\{\frac1{\sqrt2}(0,1,0,1),(1,0,0,0),\frac1{\sqrt2}(0,-1,0,1)\right\}.$$This is an orthogonal (orthonormal actually) of your space $W$.