In this paper, the stability of switched linear systems is investigated using piecewise linear Lyapunov functions. In particular, we identify classes of switching sequences that result in stable trajectories. Given a switched linear system, we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system. In the proposed approach, we assume that each individual subsystem is stable and admits a piecewise linear Lyapunov function. Based on these Lyapunov functions, we compose 'global' Lyapunov functions that guarantee stability of the switched linear system. A large class of stabilizing switching sequences for switched linear systems is characterized by computing conic partitions of the state space. The approach is applied to both discrete-time and continuous-time switched linear systems.