TITLE: Uncountable Basis?
QUESTION [20 upvotes]: I was reading up on the difference between countable and uncountable sets, and was wondering if there was a basis of uncountable size.  I now know there are, however they all seem to be covering rather high level areas in math.  So I was wondering; what is the simplest example of a basis of uncountable size for a vector space over a field?

REPLY [9 votes]: Consider $\Bbb R$ as a vector space over the rational scalars $\Bbb Q$.  As mentioned in Asaf's answer, this vector space must have a basis (at least if you accept the Axiom of Choice).  However a countable subset of $\Bbb R$ will give rise to only countably many linear combinations with rational coefficients; therefore we need an uncountable set in order to form a basis.