1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Linear Algebra Spanning Sets
Loading...