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.



Which matroids have not unique unimodular representation?
This is Corollary 10.1.4 in [Oxley](https://books.google.at/books?redir_esc=y&hl=de&id=puKta1Hdz-8C&q=10.1.4#v=snippet&q=10.1.4&f=false). Lenz gives a good account in Section 7 of [*On Powers Of Plücker Coordinates And Representability Of Arithmetic Matroids*](https://arxiv.org/pdf/1703.10520.pdf).


