Linear Algebra: Orthonormal Basis

tylerc1991

1. The problem statement, all variables and given/known data

Find an orthonormal basis for the subspace of R^4 that is spanned by the vectors: (1,0,1,0), (1,1,1,0), (1,-1,0,1), (3,4,4,-1)

3. The attempt at a solution

When I try to use the Gram-Schmidt process, I am getting (before normalization): (1,0,1,0), (0,1,0,0), (1,0,-1,2), (0,0,0,0). So obviously there is some mistake that I am making but I have checked this at least 3 times. Can someone help me and let me know if it is something on my end or the problem. Thank you.

Dick

Without have actually worked it out fully, why do you think you made a mistake? The dimension of the subspace is less than 4. You are only going to get a number of orthonormal vectors equal to the dimension of the subspace.

tylerc1991

I thought that the basis had to span R^4? And since of of the elements was (0,0,0,0) the basis can't span R^4.

Dick

It doesn't span R^4. It spans the subspace of R^4 spanned by the given vectors.

tylerc1991

I see. So will (0,0,0,0) be included in the orthonormal basis?

Dick

(0,0,0,0) is in EVERY subspace. You throw that away. It's never part of a basis. A basis is a set of linearly independent vectors. And it certainly isn't normal.