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Math Help - Sum of sets of vectors

  1. #1
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    Sum of sets of vectors

    S and T are subsets of space V

    If S and T are not subspaces S+T is not necessarily a subspace.

    However, if S and T are subspaces S+T is a subspace.

    My question is: Is it possible that S and T are not subspaces but their sum S+T is? If so, can I please see an example. If not, can you explain why not.

    Thanks
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  2. #2
    Member HappyJoe's Avatar
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    One example comes to mind:

    In \mathbb{R}, the subsets S=\{1\} and T=\{-1\} are not subspaces. But their sum S+T=\{0\} is a subspace.

    I know this is a "trivial" counterexample. I'll let you know, if I come up with a better one (if a better one exists anyway).
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  3. #3
    MHF Contributor

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    A slightly more complicated example: Let U be the subset of all vectors in R^2, <x, y> such that y= 2x and the single vector <1, 3>. The set of all <x, y> such that y= 2x is, of course, a subspace but adding <1, 3> to it, which is not in that subspace, gives a set that is not a subspace.

    Let V be the subset of all vectors in R^2 such that y= 3x [b]and[b/] the single vector <1, 2>. The set of all <x, y> such that y= 3x is a subspace but adding <1, 2>, which is not in that subspace, gives a set is not a subspace.

    Their sum, now, is the set of all vectors of the form <a+ b, 2a+ 3b>= a<1, 2>+ 3<1, b>, which is a subspace.
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