Complex Number Orthonormal Basis.

[trap101]

Find an orthonormal basis for P₂(ℂ) with respect to the inner product:

<p(x),q(x)> = p(0)q(0) + p(i)q(i) + p(2i)q(2i) the q(x) functions are suppose to be the conjugates I just don't know how to write it on the computer

Attempt:

This is where I'm having trouble. So usually I'm given a set of basis vectors, then I would apply the gram schmidt process to them. But this is throwing me for a loop. I know that the inner product of the two vectors is going to have to equate 0. But how do I choose vectors to start? I assume it wouldn't be fair to choose the standard basis vectors of P₂ and just normalize them. In fact I don't even think that would produce the necessary condition for them to be orthogonal. I'm stuck. Help.

 

[Dick]

Find an orthonormal basis for P₂(ℂ) with respect to the inner product:

<p(x),q(x)> = p(0)q(0) + p(i)q(i) + p(2i)q(2i) the q(x) functions are suppose to be the conjugates I just don't know how to write it on the computer

Attempt:

This is where I'm having trouble. So usually I'm given a set of basis vectors, then I would apply the gram schmidt process to them. But this is throwing me for a loop. I know that the inner product of the two vectors is going to have to equate 0. But how do I choose vectors to start? I assume it wouldn't be fair to choose the standard basis vectors of P₂ and just normalize them. In fact I don't even think that would produce the necessary condition for them to be orthogonal. I'm stuck. Help.

It is fair to choose the usual basis {x^2,x,1} and show they are orthogonal. Why not?