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Algebraic properites of the direct sum

  1. Sep 18, 2013 #1
    Is the following statement true? I am trying to see if I can use it as a lemma for a larger proof:

    Let ##V## be a vector space and let ##W, W_{1},W_{2}...W_{k} ## be subspaces of ##V##.
    Suppose that ## W_{1} \bigoplus W_{2} \bigoplus ... \bigoplus W_{k} = W ##
    Then is it always the case that:
    ## (W_{1} \cap W) \bigoplus (W_{2} \cap W) \bigoplus ... \bigoplus (W_{k} \cap W) = W ##

    In essence, this is asking whether there is a distributive law compatible with the set intersection operation and the direct sum operation. I am only asking for a determination of whether the statement is true for false. I will work out the proof/counterexample for myself.

    Thanks!

    BiP
     
  2. jcsd
  3. Sep 18, 2013 #2

    lavinia

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    Ask yourself whether the Wi are subspaces of W
     
  4. Sep 19, 2013 #3
    It is obvious if they are subspaces of ##W##, but what if they aren't?

    BiP
     
  5. Sep 19, 2013 #4

    lavinia

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    If W is the direct sum of the Wi then show me an element of one of the Wi's that is not in W
     
  6. Sep 20, 2013 #5

    Erland

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    As it stands, the answer is obviously yes, since each ##W_i## is a subspace of ##W##, and hence ##W_ i\cap W=W_i##.

    But I assume that there is a typo and that you meant ## W_{1} \bigoplus W_{2} \bigoplus ... \bigoplus W_{k} = V ##. Then, the answer is no. It is almost trivial to find a counterexample in R2.
     
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