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

  1. Apr 30, 2009 #1
    1. The problem statement, all variables and given/known data
    For each of the following inner product spaces V (over F) and linear transformation g:=V [tex]\rightarrow[/tex] 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>=[tex]\int_0^{1}[/tex] 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)
    2. Relevant equations

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

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

    3. 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 [tex]\overline{g(anything)}[/tex] = 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 :)
     
  2. jcsd
  3. Apr 30, 2009 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

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

    [tex]\int_0^1 e_i e_j dx[/tex]

    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>'.
     
  4. Apr 30, 2009 #3
    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?
     
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