Linear Algebra, Adjoint of a Linear Operator and Inner Product Spaces

[PrincessEmily]

Homework Statement

For each of the following inner product spaces V (over F) and linear transformation g:=V \rightarrow F, find a vector y such that g(x) = <x,y> for all x element of V.

The particular case I'm having trouble with is:

V=P₂(R), with <f,h>=\int_0^{1} f(t)h(t)dt , g(f)=f(0)+f'(1).

From the textbook Linear Algebra (fourth edition) by Friedburg and Friends.
p. 365 problem #2. c)

Homework Equations

From the theorem given in the book, I know that:
y=\sum_{i=1}^{n}\overline{g(v_i)}v_i

is a vector such that g(x)=<x,y> for all x element of V
where B={v₁,v₂...v_n} is an orthonormal basis for V.

I also know that the answer (printed in the back of the book) is y=210x² - 204x + 33 but I have no idea how to get this answer!

The Attempt at a Solution

I assigned B={1,x,x²}={e₁,e₂,e₃} as an orthornormal basis for P₂(R) (the space of polynomials of degree less than or equal to 2 over R).
Then I did:
y=g(1)e₁ + g(x)e₂ + g(x²)e₃ because the field is R, I know that \overline{g(anything)} = g(anything) so I don't worry about the conjugate of g(anything)
=(1+0)e₁ + (0+1)e₂ + (0+2)e₃
=1e₁ + 1e₂ + 2e₃
= 1 + x + 2x²

But this isn't the answer from the book... I also have no idea why the inner product was defined for me, I assumed to test my y vector?
I did I similar problem (part a) over R³ using the technique I applied above, and it worked fine.
Any help would be greatly appreciated.
Thanks :)

 

[matt grime]

If, as you state, {e_1,e_2,e_3}= {1,x, x^2} are an orthonormal basis then it would follow that

\int_0^1 e_i e_j dx

would be 0 or 1.

The inner product was given to you so that you would know what the inner product was, since there are infinitely many inner products. For example, if in your R^3 example I defined {x,y} as <x,y>/2, then you would find a different answer for the question 'find y such that g(x) = {x,y}' than 'find y such that g(x)=<x,y>'.

 

[PrincessEmily]

Thank you :)
So I should Gram-Schmidt the basis I have (and then normalize them) to find an orthonormal basis, and then do what I was doing, right?