Inner product as integral, orthonormal basis

[hocuspocus102]

Homework Statement

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.

Homework 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!

 

[Mark44]

Homework Statement

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.

Homework 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!

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.