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