Reachability analysis aims at identifying states reachable by a system within a given time horizon. This task is known to be computationally hard for hybrid systems. One of the main challenges is the handling of discrete transitions, including computation of intersections with invariants and guards. In this paper, we address this problem by proposing a state-space decomposition approach for linear hybrid systems. This approach allows us to perform most operations in low-dimensional state space, which can lead to significant performance improvements.