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Math Help - Vector subspace question

  1. #1
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    Vector subspace question

    Hi all,

    Here's a question I'm struggling with:

    Show that the set of all elements of R^2 of the form (a, -a), where a is any real number, is a subspace of R^2. Give a geometric interpretation of the subspace. Any suggestions?

    Thanks,
    Jim
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  2. #2
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    1) Does the set contain (0,0)?

    2) Is the set closed under vector addition?

    3) Is the set closed with respect to scalar multiplication?
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  3. #3
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    Quote Originally Posted by Plato View Post
    1) Does the set contain (0,0)?

    2) Is the set closed under vector addition?

    3) Is the set closed with respect to scalar multiplication?
    How do I determine if the set is closed under vector addition or scalar multiplication?
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  4. #4
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    Quote Originally Posted by Jim Newt View Post
    How do I determine if the set is closed under vector addition or scalar multiplication?
    Note: for all vectors the first and second component must be addative inverses.

    let v_1=(x,-x) and
    v_2=(y,-y) be any vectors in the subspace
    Then
    v_1+v_2=(x,-x)+(y,-y)=(x+y,-x-y)=(x+y,-(x+y))

    Since the first and 2nd components are addative inverses the same is closed under vector addition.

    I hope this helps.
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