# Finding an Orthonormal Basis

• Dec 3rd 2009, 03:56 PM
davismj
Finding an Orthonormal Basis
http://i45.tinypic.com/2mo10me.png

The inner product here is defined by the integration of p(x)q(x) from 0 to 1.

I sort of understand why this is true, but I can't prove it (so I guess I don't fully understand it).
• Dec 3rd 2009, 05:04 PM
tonio
Quote:

Originally Posted by davismj
http://i45.tinypic.com/2mo10me.png

The inner product here is defined by the integration of p(x)q(x) from 0 to 1.

I sort of understand why this is true, but I can't prove it (so I guess I don't fully understand it).

Tonio
• Dec 3rd 2009, 05:08 PM
davismj
I know how to do a Gram-Schmidt procedure. What I don't understand is how to show that the matrix for the differentiation operator on the orthonormal basis (1,sqrt(3)(2x-1),sqrt(5)(6x^2-6x+1)) is upper triangular...
• Dec 3rd 2009, 08:12 PM
tonio
Quote:

Originally Posted by davismj
I know how to do a Gram-Schmidt procedure. What I don't understand is how to show that the matrix for the differentiation operator on the orthonormal basis (1,sqrt(3)(2x-1),sqrt(5)(6x^2-6x+1)) is upper triangular...

Why did you choose precisely that basis for $P_2[x]_{\mathbb{R}}$? Check that wrt the basis $\{1,x,x^2\}$, the matrix of the diff. operator ALREADY is upper triangular, so just carry on the GS process on this basis...

Tonio