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Math Help - Subspace, span and spanning set

  1. #1
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    Subspace, span and spanning set

    I have an exam tomorrow and I can't tell the difference between the 3 definitions .

    Say V is a space over F.

    Subspace S is:
    1) a subset of V
    2) a space itself over F

    Span(S) is:
    1) a linear combination of elements in S
    2) a space itself over F

    Someone told me the other day that if S is a subspace, then S=span(s). Why? I can't tell the difference between span(S) and S.
    If span(s) a subspace, what is it's superset - S or V? Neither?

    And what is a spanning set and how it connects to span(S)?

    I'd much appreciate a concrete example, thanks
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Rita.g View Post
    I have an exam tomorrow and I can't tell the difference between the 3 definitions .

    Say V is a space over F.

    Subspace S is:
    1) a subset of V
    2) a space itself over F

    Span(S) is:
    1) a linear combination of elements in S
    2) a space itself over F

    Someone told me the other day that if S is a subspace, then S=span(s). Why? I can't tell the difference between span(S) and S.
    If span(s) a subspace, what is it's superset - S or V? Neither?

    And what is a spanning set and how it connects to span(S)?

    I'd much appreciate a concrete example, thanks
    In your second definition - the definition of span - S is not necessarily a subspace. It is just some subset of V.

    Span(S) is always a subspace, and if S is a subspace then span(S)=S holds.
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  3. #3
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    Oh, I see.
    So if I'm given a subset S of space V and I'm asked to find if it's a subspace - Is it enough to find Span(S) and show if it's equal or not to S to define S as a subspace?

    What is actually the purpose of span? What is it used for? I understand the term space and subspace, and I can't seem to connect span to any of those. The definition of span is like Latin to me heh
    I understand the words but not the sentence
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Rita.g View Post
    Oh, I see.
    So if I'm given a subset S of space V and I'm asked to find if it's a subspace - Is it enough to find Span(S) and show if it's equal or not to S to define S as a subspace?
    In essence, yes. When you show that something is a subspace you show that it is closed under some conditions and conclude that it is a subspace - you are just showing that if we take the span it is contained in the subset.

    What is actually the purpose of span? What is it used for? I understand the term space and subspace, and I can't seem to connect span to any of those. The definition of span is like Latin to me heh
    I understand the words but not the sentence
    If we take the span of the three vectors (1,0,0), (0,1,0), and (0,0,1) over \mathbb{R} we get the vector space \mathbb{R} \times \mathbb{R} \times \mathbb{R}. Can you see why?

    Span's are useful because we can write down the vector spaces in a concise manner - we do not need to write down every single element and how they interact with one another if we just write down those elements which span the space.
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  5. #5
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    Quote Originally Posted by Swlabr View Post
    In essence, yes. When you show that something is a subspace you show that it is closed under some conditions and conclude that it is a subspace - you are just showing that if we take the span it is contained in the subset.
    Just making sure - Not necessarily equal, but contained? For any subset S of V - S is a subspace if Span(s) is contained in S?

    Quote Originally Posted by Swlabr View Post
    If we take the span of the three vectors (1,0,0), (0,1,0), and (0,0,1) over \mathbb{R} we get the vector space \mathbb{R} \times \mathbb{R} \times \mathbb{R}. Can you see why?
    Because it is an independent span, therefore a basis for R3, right?

    Quote Originally Posted by Swlabr View Post
    Span's are useful because we can write down the vector spaces in a concise manner - we do not need to write down every single element and how they interact with one another if we just write down those elements which span the space.
    Ohhhh, thank you, I understand it now. Finally can make the connection to the fact that V can be infinite but spanV is finite.
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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Rita.g View Post
    Just making sure - Not necessarily equal, but contained? For any subset S of V - S is a subspace if Span(s) is contained in S?
    Yes, but it is perhaps an overly formal way of looking at it.
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