On the invariance of the set of signed vectors of a matroid.

Question

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.