I feel sorry to ask help for this problem, since I believe it's a simple one. But I go nowhere when I think alone.
The problem states :
Let be one of the following scalar product in the space of polynomials with real coefficients of degree .
a) , with different from when different from .
b) .
Prove that { } is not an orthogonal set if .
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First : I don't know how to start. Second : in the item a), I don't understand why there is the condition of the "i", if it isn't even in the equation. Maybe an error of text, so it should be ? Third : About what I must prove. It says, an orthogonal set, wouldn't it be an orthogonal basis? But even having said that, I still don't understand at all what they ask me to do. Please help me!
Yes it does. But I insist, I think it's . If the lower limit of the integral would be , then I must show that the product of 2 different functions is an odd function. It wouldn't be always true, but as you shown, if then it doesn't work. So it's not even worth looking for further n.I hope this helps.
I think I'm starting to understand the problem. So please answer me to :Now for the a). If , I get . How can they want me to say it's equal to 0? There's no way to suppose this!it says, an orthogonal set, wouldn't it be an orthogonal basis?
The set does not form a orthogonal basis for the innerproduct space with respect to the innerproduct
Look for the Orthogonality property here
Legendre polynomials - Wikipedia, the free encyclopedia
and for the Gram-Schmidt process here
Gramâ€“Schmidt process - Wikipedia, the free encyclopedia
If you use the Gram-Schmidt process on the above set you will obtain an orthogonal set with respect to the inner product mentioned it turns out to be the Legendre polynomials.