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Finding a unique vector

  1. Jun 6, 2009 #1
    1. The problem statement, all variables and given/known data
    For the given inner product space V=polynomial space of degree 2. Find a the unique vector h(x) such that <f,h>=int(f*h) from 0 to 1. and g(f)=f(0)+f'(1).



    2. Relevant equations
    Theorem: let V be a finite dimensional inner product space over F and let g:V->F be a linear transformation. Then there exists an unique vector, y in V such that g(x)=<x,y>



    3. The attempt at a solution
    well, <f,h>=g(f)

    => <f,h>=f(0)+f'(1)
    =>int(f*h) from 0 to 1=f(0)+f'(1)

    I have no clue how to solve this final step
     
  2. jcsd
  3. Jun 6, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    Be explicit. Let h(x)=h0+h1*x+h2*x^2, similarly for f(x). Now explicitly work out <f,g>=f(0)+f'(0). Hint: if the equation must be true for ALL f, then it must be true for say, f0=1, f1=0 and f2=0. Try some other choices for the coefficients of f, until you get enough equations to solve for the h's.
     
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