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Linear Algebra - adding subspaces

  1. Jan 26, 2009 #1
    1. Suppose that U is a subspace of V. What is U+U?


    2. Relevant equations:
    There's a theorem that states: Suppose that A and B are subspaces of V. Then V is the direct sum of A and B (written as A [plus with a circle around it] B) if and only if: 1) V=A+B (meaning, the two subspaces are technically able to be added), and 2) The intersection of A and B = {0}.



    3. I don't understand what the addition symbol really means in this case... I know that addition of two subspaces is NOT the same as the union of two subspaces. The union of two subspaces of V is a subspace of V if and only if one of the subspaces is contained in the other. ...Help!
     
  2. jcsd
  3. Jan 26, 2009 #2
  4. Jan 26, 2009 #3
    The sum of two subspaces A and B is the set of all sums a + b such that a is in A and b is in B. The direct sum of two subspaces A and B as subspaces of V is the set of all vectors in V that can be written uniquely as ka + jb for k,j in R (if R is the field), a in A and b in B. This is the one that is usually denoted as a circle around the plus sign.
     
  5. Jan 26, 2009 #4
    Thanks for trying to help, I think I've figured it out though:

    The definition of addition among subspaces follows this definition:

    U+U = {a+b, such that aЄU and bЄU}.

    Since U is a subspace, addition is closed in U, so adding two elements in U would simply produce another element in U.

    Therefore, U+U = U.

    A justification would be a formal proof such that U ⊆ U+U and U+U ⊆ U.
     
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