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Let $M$ be an $n \times m$ real matrix of rank $n$. We define the set of signed vectors $V(M)$ corresponding to $M$ by $V(M):=\left\{\operatorname{sign}(\vec{x}):\vec{x} \in \ker(M)\right\}$. Prove that for every $A\in GL_n$ that $V(AM)=V(M)$, i.e., that $V$ is invariant under change of coordinates.

dfeuer
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Sheila
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  • What is the sign of a vector? 2. While this problem does involve matrices, what does it have to do with matroids?
    • – vadim123 Sep 09 '13 at 04:09