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I know that dimension of $\mathbb{R} $ over $ \mathbb{Q} $ is infinite. What can i say about the cardinality of its basis mean whether it is countable or uncountable. Can we find exact basis for that.

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    $\begingroup$ The dimension is the cardinality of $\Bbb R$. You can't explicitly write a basis down though. $\endgroup$ – anon Jun 5 '15 at 2:28
  • $\begingroup$ How to show that basis is uncountable $\endgroup$ – user195218 Jun 5 '15 at 2:29
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    $\begingroup$ If a basis were countable then $\Bbb R$ would be countable. $\endgroup$ – anon Jun 5 '15 at 2:31
  • $\begingroup$ Is there any uncountable subset of real numbers which is linearly independent? $\endgroup$ – user195218 Jun 5 '15 at 2:33
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    $\begingroup$ Yes, linearly independent over the rationals $\endgroup$ – Zach Effman Jun 5 '15 at 2:34
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Note that if $(e_i)_{i \in I}\subset \Bbb R$, with $I$ countable, then: $${\rm span}_{\Bbb Q}((e_i)_{i \in I}) = \left\{ \sum_{i \in F}\alpha_ie_i \mid \alpha_i \in \Bbb Q,~F \subset I \text{ finite} \right\}$$is countable, so ${\rm span}_{\Bbb Q}((e_i)_{i \in I}) \neq \Bbb R$. So $I$ must be uncountable if it is to span whole $\Bbb R$.

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