# linear algebra proof help

• Nov 4th 2012, 02:05 PM
darts3
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
• Nov 4th 2012, 02:44 PM
girdav
Re: linear algebra proof help
a) What have you tried?

b) It's a cardinality argument.
• Nov 4th 2012, 02:47 PM
hedi
Re: linear algebra proof help
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.
• Nov 5th 2012, 03:45 PM
darts3
Re: linear algebra proof help
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?
• Nov 5th 2012, 04:28 PM
hedi
Re: linear algebra proof help
I suppose it belongs to set theory or descrete mathematics.Just use that countable union of finite sets is countable.
• Nov 5th 2012, 06:42 PM
Deveno
Re: linear algebra proof help
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.