# Abelian groups, vector spaces

1. Feb 22, 2015

### ilyas.h

1. The problem statement, all variables and given/known data
Let ℝ>0 together with multiplication denote the reals greater than zero, be an abelian group.

let (R>0)^n denote the n-fold Cartesian product of R>0 with itself.

furthermore, let a ∈ Q and b ∈ (ℝ>0)^n

we put a⊗b = $(b_1)^a + (b_2)^a + .... + (b_n)^a$

show that the abelian group (R>0)^n together with scalar multiplication

Q x (R>0)^n = (R>0)^n,
(a, b) = (a⊗b)

be a vector space over Q.

3. The attempt at a solution

proof of associativity:

p,q in Q
b in (R>0)^n

p(qb) = (pq)b

===> p(q ⊗ b) = (pq)⊗b

LHS:

p(q⊗b) = p⊗$((b_1)^q + (b_2)^q + .... + (b_n)^q)$

= $((b_1)^{pq} + (b_2)^{pq} + .... + (b_n)^{pq})$

∴associativity true.

2. Feb 22, 2015

### Dick

So far so good. What's your question?

3. Feb 22, 2015

### ilyas.h

just wanted to clarify if my logic is correct, I struggle on these sorts of Q's. Thanks.