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Linear combination subspace help

  1. Dec 14, 2007 #1
    1. The problem statement, all variables and given/known data
    In a space [tex]V^{n}[/tex] , prove that the set of all vectors
    [tex]\left\{|V^{1}_{\bot}> |V^{2}_{\bot}> |V^{3}_{\bot}> ... \right\}[/tex]
    orthogonal to any [tex]|V> \neq 0[/tex] , form a subspace [tex]V^{n-1}[/tex]
    2. Relevant equations



    3. The attempt at a solution
    I tried to make a linear combination from that set and product with <V|, I yielded nothing logical , at least I didn't understand the outcome .
    I wrote <V| as linear combination of basis in V^n vector space , I thought
    that since the |V> and those vectors share the same vector space , it might be possible that they have the same orthogonal basis (just an assumption which is probably false) .

    All it left to me the product of components of these vectors as a matrix , but as i said before I have no clue that I am doing the right thing to solve this problem .
     
  2. jcsd
  3. Dec 14, 2007 #2

    Dick

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    X is orthogonal to V if <V|X>=0. To show such vectors form a subspace you just have to show if X and Y are orthogonal to V and c is a scalar then cA and A+B are also orthogonal to V.
     
  4. Dec 14, 2007 #3
    What are A and B ?
     
  5. Dec 14, 2007 #4

    Dick

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    Ooops. I meant show cX and X+Y are orthogonal to V. Forgot my notation.
     
  6. Dec 15, 2007 #5
    how can I show it ?
    That is the problem actually .
     
  7. Dec 15, 2007 #6

    Dick

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    Use properties of the inner product! <V|(X+Y)>=<V|X>+<V|Y>, for example.
     
  8. Dec 15, 2007 #7

    HallsofIvy

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    Why in the world is this under "physics"? This is a pretty standard Linear Algebra question!
     
  9. Dec 15, 2007 #8
    I saw this problem on a quantum mechanics textbook , that's why I subscribed it in here .

    Thank you dick by the way .
     
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