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Math Help - Unions of subspaces

  1. #1
    Member Last_Singularity's Avatar
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    Unions of subspaces

    Question: Let W_1, W_2 be subspaces of vector space V. Prove that W = W_1 \cup W_2 is a subspace of V if and only if W_1 \subseteq W_2 or W_2 \subseteq W_1.

    I managed to prove it one way. I managed to show that if W_1 \subseteq W_2 or W_2 \subseteq W_1, then W_1 \cup W_2 is a subspace of V.

    However, I cannot go the other way. Surely the zero vector is contained in W because it is contained in W_1,W_2 and thus their union. So that's not the problem; the problem seems to be whether or not the structure W is closed under addition.

    So I imagine that if neither W_1 nor W_2 are subsets of each other, then there exists x \in W_1 such that x \notin W_2 and there exists y \in W_2 such that y \notin W_1. And somehow, I need to show that there exists x,y \in W such that x+y \notin W from that. This would show that if neither W_1 \subseteq W_2 nor W_2 \subseteq W_1 hold, then W=W_1 \cup W_2 fails to be a subspace of V

    How do I do so? Am I even thinking on the right track?

    Thanks!
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  2. #2
    Super Member
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    Let x \in W_1 - W_2 and let y \in W_2-W_1 consider x+y=z then if z \in W_1 then y=z-x \in W_1 which can't be, and if z \in W_2 then x=z-y \in W_2 and so W_1 \cup W_2 is not a subspace.
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  3. #3
    Member Last_Singularity's Avatar
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    Damn, owned by a one liner.

    Thanks a bunch.
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