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Math Help - Proof on vector spaces

  1. #1
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    Proof on vector spaces

    Im stuck on the following question:

    Let V be a vector space. Prove that if it is possible to find m vectors in V which are linearly independent, and n vectors which span V, then m must be less than, or equal to, n.

    Any help/hints would be great
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  2. #2
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    I posted a response but I think I read the question wrong.
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  3. #3
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    Quote Originally Posted by leftfootwonder7 View Post
    Im stuck on the following question:

    Let V be a vector space. Prove that if it is possible to find m vectors in V which are linearly independent, and n vectors which span V, then m must be less than, or equal to, n.

    Any help/hints would be great
    Call the linearly independent vectors L={v_0, v_1...v_m} and the set of n vectors which spans V S={u_0, u_2...u_n}.

    Assume m>n. Then there is at least one more vector in L than in S. L is a subset of V so must be in the span(S)=V. Since any vector in v is linearly independent, it cannot be expressed by a combination of other vectors in V, or of span(S). Even if all vectors in S are independent, the number of independent vectors in L is always greater than those in S, thus at least one vector in L is not in the span(S), which implies not in V, making a contradiction.

    Kind of messy but that's the basic idea.
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  4. #4
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    Yes i see now, proof by contradiction. Thanks
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