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Math Help - Help understanding Replacement theorem Friedberg

  1. #1
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    Help understanding Replacement theorem Friedberg

    Hello everyone, im having trouble understanding the Replacement Theorem proof outline in Friedberg Linear Algebra 4th pg 45.

    Theorem 1.10 (Replacement Theorem)

    Let V be a vector space that is generated by a set G containing exactly n vectors, and let L be a linearly independent subset of V containing exactly vectors. Then m\leq n and there exists a subset H of G containing exactly n-m vectors such that L \cup H generates V

    The proof is by induction on  m, I skipped the base case  m=0.

    Inductive step

    Let L = \{v_{1},...,v_{m+1}\}, be a linearly independent subset of V consisting of m+1 vectors. Then \{v_{1},...,v_{m}\} is linearly independent, and so we may apply the induction hypothesis to conclude that m\leq n and that theres is a subset  \{u_{1},...,v_{m-n}\} of G such that \{v_{1},...,v_{m}\} \cup  \{u_{1},...,v_{m-n}\} generates  V. Thus there exist scalars a_{1},..,a_{m},b_{1},...,b_{n-m} such that a_{1}v_{1}+...+a_{m}v_{m}+b_{1}u_{1}+...+b_{n-m}u_{n-m}=v_{m+1}.

    Note that n-m>0, lest v_{m+1} be a linear combination of \{v_{1},...,v_{m}\} which contradicts the assumption that L is linearly independent.

    I am having trouble understanding why does n-m>0.

    Could someone give me a hint so I can understand. Thanks!

    Edit: Is it beacause the induction hypothesis gives us m\leq n and and since  L has an extra vector means that they cannot be equal?
    Last edited by gordo151091; January 26th 2013 at 10:04 AM.
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  2. #2
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    Re: Help understanding Replacement theorem Friedberg

    Since m <= n, the only choices are m < n and m = n. If m = n, then from

    a_{1}v_{1}+...+a_{m}v_{m}+b_{1}u_{1}+...+b_{n-m}u_{n-m}=v_{m+1}

    we have

    a_{1}v_{1}+...+a_{m}v_{m}=v_{m+1}

    i.e., L is not linearly independent, contrary to the assumption.
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  3. #3
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    Re: Help understanding Replacement theorem Friedberg

    Thanks, I over looked n-m=0
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