a) What have you tried?
b) It's a cardinality argument.
The real numbers form a vector space over the rationals(i.e. with Q as the scalar field).
a) prove this
b) prove that this vector space has uncountable dimension
you may assume the Hamel Basis Theorem, which states that this
space has a basis.
Any help is appreciated
If a countable basis exists,every real number is a linear combination over Q of finite subset of basis element.Now use cardinals consideretions to explain why this is impossible.
for part (a) just verify that R satisfies the vector space axioms.
you can "take a short-cut" by noting that R is an abelian group under addition. then use the fact that R is a field (and thus a ring) to show that:
a(r + s) = ar + as, for any a in Q, and r,s in R.
(a+b)r = ar + br, for any a,b in Q and r in R.
a(br) = (ab)r, for any a,b in Q and r in R.
1r = r, for any real number r.