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Countability of basis

  1. Sep 28, 2008 #1
    I know that the set of real numbers over the field of rational numbers is an infinite dimensional vector space. BUT I don't quite understand why the basis of that vector space is not countable. Can someone help me???
  2. jcsd
  3. Sep 28, 2008 #2
    Suppose the basis is countable, what can you conclude from this?
  4. Sep 29, 2008 #3
    the span of a countable set is still countable? so R over Q does not have a countable basis, right?
  5. Sep 30, 2008 #4


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    If you have n basis vectors and m possible coefficients then you can only make m*n vectors. Since Q is countable, the number of possible coefficients is countable: [itex]\aleph_0[/itex]. If the number of basis vectors were also countable, the number of vectors would have to be [itex]\aleph_0\times \aleph_0= \aleph_0[/itex]: countable.
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