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Homework Help: Proof involving subsets of a vector space

  1. Jul 18, 2012 #1
    1. The problem statement, all variables and given/known data

    This is a problem from chapter 1.3 of Linear Algebra by F/I/S.

    Let [itex]W_{1}[/itex] and [itex]W_{2}[/itex] be subspaces of a vector space V. Prove that [itex]W_{1}[/itex] [itex]\cup[/itex] [itex]W_{2}[/itex] is a subspace of V iff [itex]W_{1}[/itex][itex]\subseteq[/itex][itex]W_{2}[/itex] or [itex]W_{2}[/itex] [itex]\subseteq[/itex] [itex]W_{1}[/itex].

    2. Relevant equations

    See attempt at solution.

    3. The attempt at a solution

    My proof goes as such:

    If [itex]W_{1}[/itex][itex]\subseteq[/itex][itex]W_{2}[/itex] then the union of those subspaces is [itex]W_{2}[/itex], therefore, by the given, it the union is a subspace of V.
    The same logic is used to argue the other subset.

    I'm not sure if this is correct, and additionally, I'm not sure if its a logical proof. I feel it is a little cyclical maybe. Thanks for all the help, I'm having a tough time with this text.
  2. jcsd
  3. Jul 18, 2012 #2
    Yeah that's half of the solution right there.

    Then you need to also show (it's iff = if and only if), given that [itex] W_1 \cup W_2 [/itex] is a subspace, either [itex] W_1 \subset W_2 [/itex] or [itex] W_2 \subset W_1 [/itex]
  4. Jul 18, 2012 #3
    Okay cool. Thanks. I'm self studying this book as a first exposure to linear algebra so I'm sure I'll be posting some more questions.
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