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Basis sets for V

  1. Oct 29, 2008 #1
    1. The problem statement, all variables and given/known data


    Any two basis sets for V have the same number of elements.

    2. Relevant equations

    3. The attempt at a solution

    Sounds obvious but is quite intricate to prove it.
  2. jcsd
  3. Oct 29, 2008 #2


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    Is it? I don't agree, at least if the dimension V is finite.
  4. Oct 30, 2008 #3
    You confused me even more :(
  5. Oct 30, 2008 #4


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    He said he did not agree that the proof is quite intricate.

    I know, it always confuses me when people don't agree with me, too.

    A space is said to be finite dimensional if and only if there exist a finite spanning set. In that case, since the number of vectors in a spanning set is an integer, there must exist a smallest spanning set. Since a basis is a set of vectors that is both a spanning set and independent you need to prove:

    1) The smallest spanning set is independent. (Show that if it were not a independent, you could remove one of the vectors and still have a spanning set, contradicting the fact that it is smallest.)

    2) No set of independent vectors can have more members than the smallest spanning set. (Take a supposedly independent set with more vectors and rewrite each in terms of the smallest spanning set.)
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