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Innner products and basis representation

  1. Aug 18, 2009 #1
    hi, I have a quickon vector spaces.

    Say for example we have


    X = a1U1 + a2U2 ....anUn
    this can be written as

    X = sum of ( i=0 to n) ai Ui


    now how can I get and expression of ai in therms of X and Ui.

    do we use inner product to do this...ans someone please explain how to go forward.
     
  2. jcsd
  3. Aug 18, 2009 #2

    HallsofIvy

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    If the Ui basis is "orthonormal" then, taking the inner product of X with Uk gives [itex]<X, U_n]>= a_1 <U_1,U_k>+ \cdot\cdot\cdot+ a_k<U_k,U_k>+ \cdot\cdot\cdot+ a_n<U_n, U_k>= a_1(0)+ \cdot\cdot\cdot+ a_k(1)+ \cdot\cdot\cdot+ a_n(0)= a_k[/itex].

    That is, for an orthonormal basis, [itex]a_k= <X, U_k>[/itex]. If the basis is NOT orthonormal, there is no simple formula. That's why orthonormal bases are so popular!
     
  4. Aug 18, 2009 #3
    the basis is orthonormal...so the solution you suggested should be ok...however i dont have latex and have never used it before so cant view your reply. do I just downlad latex to view the thread or do I have to do something else.
     
  5. Aug 18, 2009 #4
    thanks for the reply...as well.
     
  6. Aug 18, 2009 #5

    tiny-tim

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    LaTeX

    Hi iontail! :smile:

    You don't need to "have" LaTeX, it should be visible anyway.

    There's just something wrong with that particular LaTeX …I can't read it either :rolleyes:

    (I can't see what's wrong with the code though.)

    To see the original code, just click on the REPLY button. :wink:
     
  7. Aug 19, 2009 #6
    Here is what HallsofIvy want to write:

    [tex]<X, U_n]>= a_1 <U_1,U_k>+ \cdot\cdot\cdot+ a_k<U_k,U_k>+ \cdot\cdot\cdot+ a_n<U_n, U_k>= a_1(0)+ \cdot\cdot\cdot+ a_k(1)+ \cdot\cdot\cdot+ a_n(0)= a_k[/tex]
     
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