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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?

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1 Answer 1

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I am not sure what your question actually is, but your definition of an oriented matroid and its covectors is not correct.

Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999). Oriented Matroids. Encyclopedia of Mathematics and Its Applications 46 (2nd ed.). Cambridge University Press. ISBN 978-0-521-77750-6. Zbl 0944.52006. is an excellent reference, chapter1 would get you started and in a position to answer the question.

I have a copy of their first picture at http://oriented.sourceforge.net/examples/chapter1-6.html they label every face, edge and point in the pseudoline diagram with the corresponding covector from the oriented matroid.

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)$

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