Thread: linear algebra proof help

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

a) What have you tried?

b) It's a cardinality argument.

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.

I haven't tried anything yet, im not sure where to start. Would i have to prove that the cardinality of the Real numbers is greater than the cardinality of the rational numbers?

I suppose it belongs to set theory or descrete mathematics.Just use that countable union of finite sets is countable.

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.