What is the point in calculating orthonormal bases for R3

[seand]

I'm just learning linear algebra, where we are taking a set of basis vectors and then using gram schmidt to convert them to an orthonormal basis.

So I know I can do it. I know orthonormal is great. But as I'm modifying the basis anyway from the original, why don't I just forget the original basis and just use (0,0,1) (1,0,0) (0,1,0) and be done with it.


Is some useful information being preserved from the original basis into the orthonormal basis? I'm looking for some motivation for this process.

Thanks,

Sean

 

[Hurkyl]

The first reason that leaps to my mind is that there are other inner product spaces than Rⁿ with its standard inner product.

I could imagine, though, a situation where I would want my first basis vector to be a specific vector, and the next basis vector to lie in a specific plane, and so forth.

 

[Live2Learn]

An orthonormal basis formed using the Gram Shmidt process has a useful property. If your original basis is used as column vectors to form a matrix A, and the GS derived orthonormal basis are the column vectors of the matrix Q, then Q^T A = R. R happens to be a nice upper triangular matrix. So, I guess there is some useful information preserved when using the GS process because I can recover my original basis using A = QR.