A linear Transformation that maps the nonnegative orthant to the entire space

Question

Can we find a linear transformation $T : \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\{Tx | x \ge 0\} = \mathbb{R}^n$?

Answer 1

Suppose that a mapping $T$ with the above properties exists. Then $T$ is surjective, hence $ker(T)=\{0\}.$

Now let $w \in \mathbb R^n$ and $w \ne 0$. Then there exist unique determined $u,v \ge 0$ such that $Tu=w$ and $Tv=-w.$

This gives $u+v \in ker(T)$, hence $v=-u$. Since $u,v \ge 0$, we get $u=v=0$ and therefore $w=0$, a contradiction.