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Homework Help: Linear Algebra Spanning Sets

  1. Mar 30, 2010 #1
    1. The problem statement, all variables and given/known data

    Let V be a vector space, let p ≤ m, and let b1, . . . , bm be vectors in V such that
    A = {b1, . . . , bp} is a linearly independent set, while C = {b1, . . . , bm} is a spanning set
    for V . Prove that there exists a basis B for V such that A ⊆ B ⊆ C.


    2. Relevant equations



    3. The attempt at a solution

    I'm going on the fact that it does not mention C is linearly independent, thus by the spanning set theorem there exists a linearly independent set of vectors {bi,...,bk} which spans V. Thus, this set {bi,...,bk} is a basis for V.

    This means that the basis must at least be equal to A since B cannot be a basis for V if there is another linearly independent vecotr bp. Meaning:

    [tex] A \subseteq B [/tex]

    Also since B is a spanning set of V and is comprised of at least {b1,...,bp} it must be a subset of C since C also spans V and includes A.

    Thus

    [tex] A \subseteq B \subseteq C[/tex]
     
    Last edited: Mar 30, 2010
  2. jcsd
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