Clearly, we have a a basis for , and with a projection. is probably not an orthogonal projection, however.

If we want to normalize this, the set is a normal basis for . For ease of notation, let refer to the standard basis of . Then

where has the same definition as , other than its range. So is the projection matrix of . Please note again that this is probably not orthogonal.

However, the matrix

will never be a projection matrix for any projection of along . For consider with . Then . Since , we have , so .