1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook