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Linear Algebra: Vector Subspaces

  1. Feb 17, 2009 #1
    1. The problem statement, all variables and given/known data

    True/false: Union of two vector subspaces is a subspace.

    2. Relevant equations


    3. The attempt at a solution

    I'm unsure if this is true because I'm also unsure if it already assumes that it is closed under scalar multiplication and addition. If it is closed, then I'd like to think it is true, but it might be false if it isn't closed. Help me a lil' to know what's already assumed?
  2. jcsd
  3. Feb 17, 2009 #2
    Generally that is false, the union of two vector subspaces is not a subspace. To prove it is false, first assume that it is true and look for a counter-example. Let me see what you can come up with.
  4. Feb 17, 2009 #3
    how about span of e1 and span of e2. Then their union is not closed under vector addition, then the union is not a subspace, woah.
  5. Feb 18, 2009 #4


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    Science Advisor

    Do you not know what the "union" of two sets is? A question like this doesn't "assume" anything it doesn't say. Since a "subspace" is simply a subset that is closed under addition and scalar multiplication, surely a question that asks whether something is a subspace or not will not "assume" that it is a subspace.

    As war485 suggested, in R2, {(x, 0)} is a subspace, containing, say, (2, 0). The set {(0, y)} is a subspace containing (0, 3). Their union does NOT contain (2, 3).
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