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Encoding Matrix Rank for SAT Solvers

I’m working on a problem where I want to use a SAT solver to check that a property $P(v_1, \ldots, v_n)$ holds for a bunch of vectors $v_1, \ldots, v_n$, but I don’t care about the basis choice. In other words I want to check whether an arbitrary invertible linear transform $T$ exists so that the transformed vectors have a certain property, i.e. $P(T(v_1), \ldots, T(v_n))$. I solved this by finding an encoding for constraining the rank of a matrix. With that I can simply encode $P(M v_1, \ldots, M v_2)$ where $M$ is a square matrix constrained to have full rank and which therefore is invertible.
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