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Math Help - Prove Linearity of the Map

  1. #1
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    Prove Linearity of the Map

    Prove the linearity of the map T : (R(>0), +*, **) -->(R, +, *), a --> ln(a) where R carries the standard vector space structure and R(>0) has the vector structure a +* b = ab and a ** b = a^b.

    To prove linearity, you have to prove that T(x + y) = T(x) + T(y) and T(cx) = cT(x), but I don't understand what to do with the different vector spaces.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    (i)

    \forall x,\forall y\in \mathbb{R}^+-\{0\}:\;\;T(x+^*y)=T(xy)=\ln (xy)=\ln x+\ln y=T(x)+T(y)

    Try:

    (ii)


    \forall \lambda,\forall x\in \mathbb{R}^+-\{0\}:\;\;T(\lambda^{\;**}x)=\ldots=\lambda T(x)


    Fernando Revilla
    Last edited by FernandoRevilla; February 21st 2011 at 01:18 AM.
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