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Math Help - Inner product problem

  1. #1
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    Inner product problem

    Let V be an inner product space, and let W be a finite-dimensional subjspace of V. If x is not in W, prove that there exist a vector y in V such that y is in the orthogonal complement of W, but <x,y> is not equal to 0.

    Proof. By a theorem, there exist a unique vector u in W and z in orthogonal complement of W such that x = u + z.

    <x,z> = <u+z,z> = <u,z> + <z,z>. Now, z is not 0, therefore <z,z> cannot equal to 0, so z is the vector y that we are looking for.

    Is that right?
    Last edited by tttcomrader; April 2nd 2008 at 01:36 PM.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let V be an inner product space, and let W be a finite-dimensional subjspace of V. If x is not in W, prove that there exist a vector y in V such that y is in the orthogonal complement of W, but <x,y> is not equal to 0.

    Proof. By a theorem, there exist a unique vector u in W and z in orthogonal complement of W such that x = u + z.

    <x,z> = <u+z,z> = <u,z> + <z,z>. Now, z is not 0, therefore <z,z> cannot equal to 0, so z is the vector y that we are looking for.

    Is that right?
    Yes!
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