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Lineaer algebra - orthogonality

  1. Feb 8, 2013 #1
    1. The problem statement, all variables and given/known data
    Let<x,y> be an inner product on a vector space V, and let e1, e2,...,en be an orthonormal basis for V.
    Prove: <x,y> = <x,e1><y,e1>+...+<x,en><y,en>.

    2. Relevant equations

    <x,x> = abs(x)
    a<x,y> = <ax,y> = <x,ay>

    3. The attempt at a solution

    RHS
    =<x,y>
    = x1y1+...+xnyn
    = x1<e1,e1>y1<e1,e1>+....+xn<en,en>yn<en,en>
    * <e,e>=1
    =e12<x1,e1>+...+en2<xn,en>

    And i cannot find a way to take the e2 out of the equation.
    Any help is appreciated
     
  2. jcsd
  3. Feb 8, 2013 #2

    ehild

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    How is an orthonormal basis defined?

    A subset {v_1,...,v_k} of a vector space V, with the inner product <,>, is called orthonormal if <v_i,v_j>=0 when i≠j. That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: <v_i,v_i>=1.

    ehild
     
  4. Feb 8, 2013 #3
    Orthonormal basis
    A set of vectors for an inner product space whose are linearly independence and span V and orthonormal.

    orthonormal: orthogonal (<x,y>=0) and each vector has length one.
     
  5. Feb 8, 2013 #4

    HallsofIvy

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    Under "relevant equations" you have "<x, x>= abs(x)". There is NO "x2" so what, exactly do you mean by "en2"? <en, en>?
    You also say "orthonormal: orthogonal (<x,y>=0) and each vector has length one."
     
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