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Math Help - subset of a vector space

  1. #1
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    subset of a vector space

    how do you prove this theorem?

    theorem:
    let S be a subset of a vector space V
    (a) Then span S is a subspace of V which contains S
    (b) if W is a subspace of V containing S, then span s ⊆ W

    i would think that,

    assume v, w ∈ span S, where
    v= a1va + a2v2+...+ amvm and w=b1w1+b2w2+...bmwm
    then

    v+w=a1v1+..+amvm+b1w1+...bmwm.
    thus span S is a linear combination of the vectors in S.

    does that show that it is a subspace of V? it seems to me that i have just proven that span S is a vector space....

    then how do i do the second part?

    thanks!
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    how do you prove this theorem?

    theorem:
    let S be a subset of a vector space V
    (a) Then span S is a subspace of V which contains S
    (b) if W is a subspace of V containing S, then span s ⊆ W

    i would think that,

    assume v, w ∈ span S, where
    v= a1va + a2v2+...+ amvm and w=b1w1+b2w2+...bmwm
    then

    v+w=a1v1+..+amvm+b1w1+...bmwm.
    thus span S is a linear combination of the vectors in S.

    does that show that it is a subspace of V? it seems to me that i have just proven that span S is a vector space....

    then how do i do the second part?

    thanks!
    Hi
    For part 1. Please show that if  x\in span(S) => x\in V
    You will be done (This will prove span is a subset)

    You have already done it in part one. Just replace V with W - the question is similar to part 1 above. Here you jut have to prove span S is a subset of W. A stronger inference is that it is a sub-space of W as well
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  3. #3
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    sorry!

    could you explain further becos i cant understand what you mean..
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  4. #4
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    Quote Originally Posted by alexandrabel90 View Post
    how do you prove this theorem?

    theorem:
    let S be a subset of a vector space V
    (a) Then span S is a subspace of V which contains S
    (b) if W is a subspace of V containing S, then span s ⊆ W

    i would think that,

    assume v, w ∈ span S, where
    v= a1va + a2v2+...+ amvm and w=b1w1+b2w2+...bmwm
    then

    v+w=a1v1+..+amvm+b1w1+...bmwm.
    thus span S is a linear combination of the vectors in S.
    Be more careful here: the span of S is not a linear combination of the vectors of S. Instead it is the set of all linear combinations of vectors from S.

    does that show that it is a subspace of V? it seems to me that i have just proven that span S is a vector space....
    First, before attempting to prove any proposition you have to be very clear about how the terms in that proposition have been defined.
    Because the span of a set S with respect to a vector space V can be defined in different ways, you may or may not have been successful in your attempt. (I can think of at least three ways to define the span of S: first, as the set of all linear combinations of S; second, as the smallest subspace of V containing S; and, third, as the intersection of all subspaces of V containing S.)
    Basically, you need to show, basing yourself on the definition of span S that you have been given (somwhere), that that set of vectors satisfies all the axioms of a vector space.


    then how do i do the second part?

    thanks!
    Surely, if S is a subset of W and you are forming a linear combination of elements of S (and thus of W), you get another element of W, because W is a vector (sub-)space: vector (sub-)spaces are, by definition, closed under the operation of forming linear combinations of some of their elements.
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