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    Vector Subspaces

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    Suppose that V and W are vector subspaces of Rn. If I define:

    V + W = {v+w: v belonging to V, w belonging to W}

    How can I prove that V+W is also a vector subspace of Rn and ALSO how could verify that (for example)
    <(1,0,1), (-1,0,1)> + <(3,2,1)> = R3


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    Follow up from OTA on Student Comment:
    To answer your questions in the comment:

    Yes, it works with any square matrix.

    I have in my old algebra text a theorem (Elementary Linear Algebra with Applications by W. Keith Nicholson):
    The ...

    Solution Summary

    Vector subspaces are investigated. The solution is detailed and well presented.