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Math Help - Subsets generating supspaces

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    Subsets generating supspaces

    Show that two subsets A and B of a vector space V generate the same subspace if and only if each vector in A is a linear combination of vectors in B and vice versa.

    (a) if two subsets A and B of a vector space V generate the same subspace, then each vector in A is a lin. comb. of vectors in B.
    I am thinking by contradiction: Suppose subsets A and B of a vector space V don't generate the same subspace and a in A is a lin. comb. of B.

    \mathbf{a}_i=\sum_{i\in I}\lambda_i\mathbf{b}_i, \ \lambda_i\in\mathbb{F}
    From this, it seems trivial that A generates the same subspace. However, I could be wrong or I just don't know how to articulate it from here

    Don't worry about answering the other direction. I am focusing on this one now.
    Last edited by dwsmith; August 26th 2011 at 04:20 PM. Reason: + am
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    MHF Contributor Drexel28's Avatar
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    Re: Subsets generating supspaces

    Quote Originally Posted by dwsmith View Post
    Show that two subsets A and B of a vector space V generate the same subspace if and only if each vector in A is a linear combination of vectors in B and vice versa.

    (a) if two subsets A and B of a vector space V generate the same subspace, then each vector in A is a lin. comb. of vectors in B.
    I am thinking by contradiction: Suppose subsets A and B of a vector space V don't generate the same subspace and a in A is a lin. comb. of B.

    \mathbf{a}_i=\sum_{i\in I}\lambda_i\mathbf{b}_i, \ \lambda_i\in\mathbb{F}
    From this, it seems trivial that A generates the same subspace. However, I could be wrong or I just don't know how to articulate it from here

    Don't worry about answering the other direction. I focusing on this one now.
    I think you're making it more difficult than it is. Suppose that \text{span }A=\text{span }B, then in particular, since A\subseteq\text{span }A we of course have that A\subseteq\text{span }B which by definition means each element of A is a linear combination of elements of B, etc.
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    Re: Subsets generating supspaces

    Quote Originally Posted by Drexel28 View Post
    I think you're making it more difficult than it is. Suppose that \text{span }A=\text{span }B, then in particular, since A\subseteq\text{span }A we of course have that A\subseteq\text{span }B which by definition means each element of A is a linear combination of elements of B, etc.
    How can we go from subsets A and B to speaking about their spans?
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    MHF Contributor Drexel28's Avatar
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    Re: Subsets generating supspaces

    Quote Originally Posted by dwsmith View Post
    How can we go from subsets A and B to speaking about their spans?
    Saying that they generate the same subspace is equivalent to saying they have equal spans.
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    Re: Subsets generating supspaces

    Then for the other direction assuming the contrary would be the way to go? Since we are assuming the contrary, there exists an a in A s.t. a\neq\sum\lambda_ib_i. Therefore, the span A > span B which is a contradiction.

    Correct?
    Last edited by dwsmith; August 27th 2011 at 01:05 PM.
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