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Math Help - isomorphism question

  1. #1
    Newbie
    Joined
    Apr 2009
    Posts
    1

    isomorphism question

    a belongs to R
    show that the map
    L: R^n------R^n>0
    (R^n>0 denote the n-fold cartesian product of R>0 with itself)
    (a1)
    (....) ----------
    (an)

    (e^a1)
    (.....)
    (e^an)
    is a isomorphism between the vector space R^n and the vector space R^n>0
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  2. #2
    Super Member Gamma's Avatar
    Joined
    Dec 2008
    From
    Iowa City, IA
    Posts
    517

    Not sure I understand

    you have \phi : \mathbb{R}^n \rightarrow \mathbb{R}_+^{n} by \phi(a_1,...,a_n)=(e^{a_1},...,e^{a_n})?

    This is bijective because \phi ^{-1} ((a_1, ..., a_n))= (ln(a_1),..., ln(a_n))

    I think you can check it is a morphism.
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