Given a $m \times n$ matrix $A$ with $m>n$, I would like to enumerate all possible sign patterns $w$ generated by $Av$ for all $v \in \mathbb{R}^n$. More specifically, if $(Av)_i \geq 0$ then $w_i = 1$, otherwise $w_i=0$. How can I do this?
I believe reframing this question as the number of quadrants than an $n$-dimensional subspace in $\mathbb{R}^m$ passes through allows us to apply the results from this question. In other words $2\sum_{i=0}^{n-1}{m-1 \choose i}$ is an upper bound on the number of unique $w$.
I believe this question is related to oriented matroids but I am not familiar enough with matroid theory at the moment to make any further assertions.
11/26/23: Cross posting my Math SE question which was asked last week. There is also a bounty of +50 open on Math SE that expires in 3 days. Feel free to respond here and link your response in an answer on Math SE if you are interested the bounty. Will consider a bounty here afterwards.
