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Just randomly making up some subspaces

  1. Sep 20, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove or give a counterexample: if U1, U2, W are subspaces of V such that:

    U1 + W = U2 + W then U1 = U2


    2. Relevant equations



    3. The attempt at a solution
    I would be inclined to say that it's true, however I took a peek at the back of the book and that's incorrect. Here's why I thought it was correct.

    U1 = {(x,y) ∈ F2: y = x}
    W = {(x,y) ∈ F2: x ∈ F, y ∈ F}
    (just randomly making up some subspaces)

    U1 + W = W + something
    then that something has to be U1 = U2

    I can't think of any U1 that would change if I add W to it.

    Does anyone have any suggestions?
     
  2. jcsd
  3. Sep 20, 2009 #2

    Office_Shredder

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    Re: Subspaces

    Ok, but you can't just pick U1 and W and demonstrate it's true for only that case. What if W=V?
     
  4. Sep 20, 2009 #3

    HallsofIvy

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    Re: Subspaces

    Suppose V= [itex]R^2[/itex], [itex]U_1= \{k\vec{i}\}[/itex] for k a real number, [itex]U_2= R^2[/itex] and W= [itex]R^2[/itex]. What are U1+ W and U2+ W?
     
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