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Math Help - Generalizing the Replacement Theorem

  1. #1
    Senior Member I-Think's Avatar
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    Generalizing the Replacement Theorem

    Let \beta be a basis for a vector space V and let S be a linear independent subset of V. Prove there exists a subset of \beta, S_1, such that S\cup{S_1} is a basis for V

    Proof
    Let S_1 be the set of all elements in \beta such that S\union{S_1} is linearly independent. Now consider S\cup{S_1}

    Note that every vector in S can be expressed as a linear combination of vectors in \beta. So it follows that every vector in V can be expressed as a linear combination of vectors in S\cup{S_1}
    Thus S\cup{S_1} is linearly independent and generates V, so it is a basis.

    Is this proof 100% correct?
    Last edited by I-Think; June 5th 2011 at 05:41 PM.
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  2. #2
    Senior Member Tinyboss's Avatar
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    Can you clarify the first line after "proof"? I'm not sure if SS_1 is meant to be S\cup S_1, or what. Also, (and I'm going with a different interpretation here) it's entirely possible (in fact overwhelmingly likely) that every element of B is linearly independent (I would say "not in the span of") S. In that case, you will get a set that's too big to be a basis, at least in finite dimension. Even in infinite dimension, it won't be linearly independent.
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