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Math Help - spanning

  1. #1
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    spanning

    how do i prove that every subspace is a spanning set for itself?


    sorry im very weak at proofing.

    my working:

    let S be the subspace.
    so we need to show that S = span(S)

    we shall show that S⊂span S ( shown in previous post) and span S ⊂S

    to show that spanS ⊂ S,

    lets assume that S is in n dimension where xi∈ S where 1<i<n

    span S =a1x1+...+anxn ⊂ S as S being a subspace obeys the laws of scalar multiplication and addition.


    is this done like this?
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  2. #2
    Member HappyJoe's Avatar
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    Your wording seems a bit off to me, but your idea seems quite right.
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  3. #3
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    how should i change my explanation then?

    thanksss!
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  4. #4
    Member HappyJoe's Avatar
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    You want to show that when S is a subspace, we have that span S is contained in S. We don't really need to know anything about the dimension of S.

    Take an element y in span S. Since span S consists of finite linear combinations of elements from S, we can write y = a_1x_1+...+a_nx_n for some elements x_i in S, and some scalars a_i (note: this n need not coincide with the dimension of S). And then, as you say, since S is closed under scaling and under vector addition, then a_1x_1+...+a_nx_n will also be in S, whereas y will be in S. So all elements of span S are contained in S - hence span S is a subset of S.
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  5. #5
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    If v is in S then v= 1v. QED

    (You posted this same question 10 hours earlier and got this same answer from Ackbeet, thanked him and said you understood. What went wrong?)
    Last edited by HallsofIvy; October 3rd 2010 at 04:59 AM.
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  6. #6
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    that previous question that i think you are referring to is for S ⊂ span(S) which i stated in my question here that i know how to solve that. now im trying span(S) ⊂ S
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