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Matroid $M$ is represented by real vectors, and we know that any base of $M$ generates the same lattice (this is called unimodular representation, I guess.) If we change the sign of any vector, we again get a representation of the same type. Are there examples when two unimodular representations of $M$ are not equivalent under changes of signs and linear transforms? In other words, two systems of vectors $(e_1,\dots,e_n)$, $(f_1,\dots,f_n)$ are unimodular, define the same matroid on $\{1,\dots,n\}$, but there do not exist linear map $T$ and signs such that $f_i=\pm Te_i$ for all $i$.

Upd. It looks like there is no counterexample, i.e., unimodular representation is always essentially unique. Thus my question is a reference to this result.

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This is Corollary 10.1.4 in Oxley. Lenz gives a good account in Section 7 of On Powers Of Plücker Coordinates And Representability Of Arithmetic Matroids.

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