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Orthogonal Complements.

  1. Apr 23, 2009 #1
    Hi, I was just reading about Orthogonal complements.

    I managed to prove that if V was a vector space, and W was a subspace of V, then it implied that the orthogonal complement of W was also a subspace of V.

    I then proved that the intersection of W and its orthogonal complement equals 0.

    However, I am wondering if the union of W and its orthogonal complement equals V?

    Can anyone please answer that, and if so, can you give a proof?

    Thanks.

    -xfunctionx-
     
  2. jcsd
  3. Apr 23, 2009 #2

    CompuChip

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    It is true, says see this page. The links on the page will give you some hints as to in which direction the proof should be found.
     
  4. Apr 23, 2009 #3

    matt grime

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    The union is not V: the union of two vector subspaces is not in general a subspace: just remember that R^2 is not the union of two lines.

    V is the vector space sum of W and its complement.
     
  5. Apr 23, 2009 #4
    As Matt Grime said, the union is not V. The union would not even be a subspace of V, unless W = {0} or W = V. However, the direct sum of W and its orthogonal complement is equal to V.
     
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