Real Numbers Vector Space: Countability of Basis

leon8179
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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?
 
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Suppose the basis is countable, what can you conclude from this?
 
the span of a countable set is still countable? so R over Q does not have a countable basis, right?
 
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: \aleph_0. If the number of basis vectors were also countable, the number of vectors would have to be \aleph_0\times \aleph_0= \aleph_0: countable.
 

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