How to demonstrate a set is a real vector space (set governed by nonstandard operations) I am really not that familiar with questions that ask you to work with a operation vector space, even less with the English terms for it. I am... quite lost. How would you prove that it is a real vector space? Furthermore, is the null vector element just a $(1_1,1_2,...,1_n)$ vector? It also asks for a opposite element to each vector... I suppose that would be $u=\{\lambda v\in \Bbb R^n_+\mid\lambda=-1\}$ since $-v\oplus v=\varnothing$    ... am I even close to understanding?
Below is the given set, all real positives:

$\Bbb R^n_+=\{x_1,x_2,x_3,...,x_n\mid x_1>0,x_2>0,...,x_n>0\}$

The question sorta says, and I remind you I am translating this, "Show that the set that is given with the indicated operations, is a real vector space."
Operations:

$(x_1,...,x_n)\oplus(y_1,...,y_n)=(x_1y_1,...,x_ny_n)$

and the scalar multiplication:

$\lambda \otimes (x_1,...,x_n)=\left(x^\lambda_1,...,x^\lambda_n\right)$

 A: You're not quite on target with your interpretation of the "additive" inverse of a given, arbitrary $u=(u_1,...,u_n)\in\Bbb R^n_+.$ You are correct about the "additive" identity. Use the latter together with the definition of "addition" to deduce what $\ominus(u_1,...,u_n)$ must be. It may help you to recall/note/prove that $-1\otimes(u_1,...,u_n)$ is what you're looking for.
As for what you must do, the idea is to check that the vector space axioms hold for the "addition" operation and the "scalar multiplication" operation thus defined. These should follow fairly readily from properties of arithmetic on positive real numbers. Let me know if you get stuck on any of them, or if you just want someone to check your reasoning.
It might help you (with intuition, and possibly with proof) to notice that $\Bbb R^n_+$ is the image of $\Bbb R^n$ under the map $$(x_1,...,x_n)\mapsto\left(e^{x_1},...,e^{x_n}\right),$$ that this map is injective, and that this map is "operation-preserving" (that is, "addition" in $\Bbb R^n_+$ looks like the image of usual addition in $\Bbb R^n$ under the mapping, and "scalar multiplication" in $\Bbb R^n_+$ looks like the image of usual scalar multiplication in $\Bbb R^n$ under the mapping).
