Linear Algebra: Orthonormal Basis

[tylerc1991]

Homework Statement

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)

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]

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.

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]

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.

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

 

[tylerc1991]

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

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

 

[Dick]

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

(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.