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Uncountable Basis

It can be shown that R (the set of all real numbers) is an infinite-dimensional vector space over Q (field of rationals).
Is it true that any basis (by basis I mean algebraic basis or Hamel basis) of R over Q has to be uncountable ?

Solution Preview

Yes. It will have to have the same cardinality as R, in fact.
Let B be a basis of R over the field Q.
We know that every element of R can be ...

Solution Summary

Uncountability of a basis is investigated.