# isomorphism question

• April 22nd 2009, 12:55 PM
ak123456
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
• April 22nd 2009, 02:19 PM
Gamma
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.