# Inner product as integral, orthonormal basis

1. May 5, 2010

### hocuspocus102

1. The problem statement, all variables and given/known data

Define an inner product on P2 by <f,g> = integral from 0 to 1 of f(x)g(x)dx. find an orthonormal basis of P2 with respect to this inner product.

2. Relevant equations

So this is a practice problem and it gives me the answer I just don't understand where it came from. It says, "We first find a basis of P2 then use Gram-Schmidt to create an orthonormal basis. Fix a basis B = {f1(x) = 1, f2(x) = x - 1/2, f3(x) = (x-1/2)2}." then it goes on to use Gram-Schmidt which I understand. I just don't get where the basis came from, if anyone can explain. Thanks!

2. May 5, 2010

### Staff: Mentor

The standard basis for P2 would be {1, x, x2}, but there are many possible bases, and they just came up with a different one. As long as there are three functions that are linearly independent, the set is a basis, so where it came from shouldn't be a concern. Just take it as a given, and proceed with Gram-Schmidt to find an orthogonal basis, and then normalize each function.