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Linear Algebra - Infinite fields and vector spaces with infinite vectors

  1. Jan 23, 2013 #1
    1. The problem statement, all variables and given/known data
    Let F be an infinite field (that is, a field with an infinite number of elements) and let V be a nontrivial vector space over F. Prove that V contains infinitely many vectors.


    2. Relevant equations
    The axioms for fields and vector spaces.


    3. The attempt at a solution
    I'm thinking this is easier than I'm making it. Can I say, at the very least, F is countably infinite, so then there exist an infinite amount of scalars to apply to V?
     
  2. jcsd
  3. Jan 23, 2013 #2

    HallsofIvy

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    Staff Emeritus
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    Yes, it really is that easy. Since V is a non-trivial vector space it contains a non-zero vector, v. And then for any a in F, av is in V. The "non-trivial" part of the proof is showing that if [itex]a_1\ne a_2[/itex] then [itex]a_1v\ne a_2v[/itex] but that is easy to show.
     
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