# Basic feasible solution of $P=\{x\in\mathbb{R}^n|Ax=b,x\ge0\}$

Has the subset of $$\mathbb{R}$$, defined as $$P=\{x\in\mathbb{R}^n|Ax=b,x\ge0\}$$, a basic feasible solution?

I think that no, because maybe P=$$\emptyset$$, so an empty polyhedron does not have basic solution. Am I wrong?

In the case $$P\neq\emptyset$$, then it's sure that P has a basic feasible solution?

By definition, a basic feasible solution (BFS from now on) must satisfy both $$Ax=b$$ and $$x\ge0$$. Therefore, if $$P = \emptyset$$, P has no BFS. However, by weakening the hypothesis one can show that $$P$$ will always have at a least one BFS.
If we let $$A \in \mathbb{R}^{m \times n}$$, rank $$A = m$$, $$P = \{x \in \mathbb{R}^n|Ax=b, x\ge0\}$$, $$P \ne \emptyset$$, then there exists at least one BFS.