Consider a linear system $\newcommand{\vv}{\mathbf{v}}\newcommand{\xx}{\mathbf{x}}\newcommand{\bb}{\mathbf{b}}A\xx=\bb$ with $\xx\in\mathbb R^m$, $\mathbf b\in\mathbb R^n$, and suppose $\operatorname{rank}(A)=n$.
We then say that $\xx_B\in\mathbb R^m$ is a Basic Feasible Solutions (BFSs) if $A\xx_B=\bb$, at least $m-n$ components of $\xx_B$ are zero, and $\xx_B\ge0$.
On the other hand, the set of feasible solutions $P\subseteq\mathbb R^m$ is the convex polyhedron of all $\mathbf x\ge0$ such that $A\xx=\bb$.
It is mentioned e.g. here that BFSs correspond to the extreme points of $P$. Is this a one-to-one relation? Are BFSs all and only the extremes of $P$? What's a good way to prove this?
