Problem:

Find a basis for , where

Attempt:

First I tried taking an arbitrary 4x1 matrix, and dotting it with a vector from the span of W. Then I realized that it must be orthogonal to all of the vectors in the span of W, so I trashed that idea.

I know that is the kernel of the orthogonal projection onto W. So therefore,

This is where I get lost, and I'm not sure how to continue or even if I'm on the right track. Any help is greatly appreciated.

Here is my work: