(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find an orthonormal basis for P2(R) using the Gram-Schmidt orthogonalization process, with the inner product defined by <f,g> = integral f(t)g(t) dt from 0 to 1. Then, if T(f) = f '' (1) + x*f (0), find T*(f).

2. Relevant equations

Given a basis a = {w1, w2, ... , wn}, we compute the orthonormal basis B = {v1, v2,...,vn} by Gram-Schmidt:

v1=w1

v2= w2 - (<w2, v1>/(<v1,v1>^2))*v1

v3 = w3 - (<w3, v1>/(<v1,v1>^2))*v1 - <w3, v2>/(<v2,v2>^2))*v2

3. The attempt at a solution

I just need someone to verify this and tell me if I'm right. I'm a bit confused at how this is so much more complex then the case when the integral is from -1 to 1 (that will just give you the Legendre polynomials and I'm able to compute them just fine). When I used the standard ordered basis {1,x,x^2} with Gram Schmidt, I got this:

B = {1, (3^1/2)*2*(x - 1/2), (12/1009)(x^2 - (9/4)x - 1/3)*(5045)^1/2}

which is REALLY ugly. I don't think this works, as I keep trying to take the inner product of v1 and v2, but I don't get zero... but I also don't see any mistakes in my work, so I don't know.

If I could just get this orthonormal basis, then I know I just need to get the matrix representation of T which would be really easy, and then take its transpose. From that point, I'm not sure how to get from the matrix [T*]B back into an expression T*(f).

Please help! Even if someone just knows what the correct orthonormal basis for this inner product is.

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# Help with gram schmidt and T*!

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