Consider a rectangle matrix $A\in\mathbb{R}^{m\times n}$ with $m\ge n$, and the set of all vectors $x\in\mathbb{R}^n$ satisfying $Ax\ge\vec{0}$. I note this set is closed under multiplication by a scalar and linear combinations with positive coefficients.
Can one define for it the analogue of a basis of linear sub-space? Is there some notion of well-defined dimension for such a set?
EDITED: Thanks to @AlgebraicPavel we know it is a convex cone, or the dual cone of A.