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Abelian groups, vector spaces

  1. Feb 22, 2015 #1
    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 = [itex](b_1)^a + (b_2)^a + .... + (b_n)^a[/itex]

    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⊗[itex]((b_1)^q + (b_2)^q + .... + (b_n)^q)[/itex]

    = [itex]((b_1)^{pq} + (b_2)^{pq} + .... + (b_n)^{pq})[/itex]

    ∴associativity true.
     
  2. jcsd
  3. Feb 22, 2015 #2

    Dick

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    Science Advisor
    Homework Helper

    So far so good. What's your question?
     
  4. Feb 22, 2015 #3
    just wanted to clarify if my logic is correct, I struggle on these sorts of Q's. Thanks.
     
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