We consider discrete-time systems with linear dynamics and a finite input alphabet. We propose a sufficient condition for the existence of finite uniform bisimulations, and an algorithm that terminates when the condition holds and constructs the corresponding equivalence classes. We derive a stronger sufficient condition for the construction of arbitrarily many equivalence classes, together with a corresponding constructive algorithm. Finally, we derive a necessary condition for the existence of finite uniform bisimulations. All our conditions are topological in nature.