Real Numbers Vector Space: Countability of Basis

[leon8179]

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?

 

[d_leet]

Suppose the basis is countable, what can you conclude from this?

 

[leon8179]

the span of a countable set is still countable? so R over Q does not have a countable basis, right?

 

[HallsofIvy]

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.