The time scale theory is introduced to study the stability of a class of switched linear systems on non-uniform time domains, where the dynamical system commutes between a continuous-time linear subsystem and a discrete-time linear subsystem during a certain period of time. Without assuming that matrices are pairwise commuting, some conditions are derived to guarantee the exponential stability of the switched system by considering that the continuous-time subsystem is stable and the discrete-time subsystem can be stable or unstable. Some examples show the effectiveness of the proposed scheme.