Consider the vector space defined hereabove. It may be proven of $(\mathbb{R^+})^n$, that it is isomorphic with $(\mathbb{R})^n$ and so possesses a dimension of $n$. I wish to know how a basis (which shall be of size $n$) may be obtained for this vector space, provided the operations of vector addition and scalar multiplication are not standardly defined.
Best Answer
An isomorphism $\Psi$ from $\Bbb R^n$ onto $(\Bbb R^+)^n$ is$$(x_1,\ldots,x_n)\mapsto\left(e^{x_1},e^{x_2},\ldots,e^{x_n}\right).$$So, if $\{e_1,e_2,\ldots,e_n\}$ is the standard basis of $\Bbb R^n$ (or any other basis, for that matter), $\left\{\Psi(e_1),\Psi(e_2),\ldots\Psi(e_n)\right\}$ is a basis of $(\Bbb R^+)^n$.