What is the orthonormal basis for P2(R) using Gram-Schmidt and T*?

[quasar_4]

Homework Statement

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).

Homework 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

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.

 

[Dick]

Your version of Gram-Schmidt doesn't look quite like I know it. Don't you subtract all of the projections of the previous vectors and then normalize the result (as in the wikipedia exposition)? The first and second vectors in your results look fine, but the third one is way off.

 

[quasar_4]

aha, I see now, I was squaring a norm when I shouldn't have been... it seems to work fine now! Finally... thanks!

 

[moo5003]

As for finding T in matrix form instead of finding an orthonormal basis you could always just see how T acts on ax^2 +bx + c and then correlate that with a 3-tuple.

Then you just easily extrapolate a 3x3 matrix wrt the standard basis on R_3 (Which is orthonormal anyways).