Are these theorems equivalent (possible notation variation)?
Let $A$ be an $m\times n$. Exactly one of the following system has a solution:
1)$Ax>0$
2)$A^ty=0,y\ge 0,y\neq 0$
And
Exactly one of the following system has a solution:
1) $y^TA>0,$ for some $y\in\mathbb R^n$
2)$Ax=0,x\ge 0$ for some non-zero $x\in\mathbb R^m$
??