# Oriented matroids with support on a complement

This is an oriented matroids question, so if there is anyone who is familiar with this area of study, I would greatly appreciate your help.

Context

Suppose $$M$$ is an $$n\times m$$ real matrix of rank $$n$$. The set $$V^*(M)$$ is the set of $$\textbf{signed covectors}$$ corresponding to $$M$$ and is defined as $$V^*(M):=\left\{sign(\textbf{x}) : \textbf{x}\in \text{row}\hspace{1mm} M\right\}$$.

(The $$\textit{sign}$$ of a vector is simply the sign of its components, so for example if the vector is $$(1,-2,0)$$ its sign vector is $$(+,-,0)$$).

The set of $$\textbf{signed cocircuits}$$ corresponding to $$M$$ is $$C^*(M)$$ and its elements are the minimal nonzero elements of $$V^*(M)$$.

Trying to prove the statement

For every $$D\in C^*(M)$$ there exists an independent set $$\left\{ v_{i_1},\ldots ,v_{i_{n-1}}\right\}$$ such that $$D$$ is supported on the complement of $$\left\{i_1,\ldots ,i_{n-1}\right\}$$.

First of all, I would like to know what this question is exactly asking. I am completely new to this subject and am struggling very much with it. Second of all, I was given two clues.

Clues

1. Reduce to rank 2 and draw a picture.

2. Ponder reduced-row-echelon form.

How can I prove the statement?

Their matrix from which that oriented matroid can be derived is: $\left( \begin{array}{ccc} 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 \end{array} \right)$