Determinantal conditions for homomorphic sensing

With $k$ an infinite field and $τ_1,τ_2$ endomorphisms of $k^m$, we provide a dimension bound on an open locus of a determinantal scheme, under which, for a general subspace $V \subseteq k^m$ of dimension $n \le m/2$, for $v_1,v_2 \in V$ we have $τ_1(v_1)=τ_2(v_2)$ only if $v_1=v_2$. Specializing to permutations composed by coordinate projections, we obtain an abstract proof of the unlabeled sensing theorem.