Several key management schemes for access control in a user hierarchy have been proposed in the recent years. However, most of them have dealt with key management issues for tree structured hierarchies, which result overheads when they are applied to the linear hierarchies. Linear hierarchy comprises a particularly interesting type of hierarchies, and it appears in a wide range of applications such as secure communications within security corporations and multi-layered data streaming. In this paper, we propose an effective key management scheme for linear hierarchies. Compared to the recently proposed Hassen et al.'s scheme, our scheme reduces the storage overhead of each class significantly. Further, use of symmetric-key cryptosystem (Advanced Encryption Standard algorithm) makes the key sizes for security classes to reduce in our scheme. In addition, our scheme is secure against possible attacks required for a linear hierarchy access control scheme. Security along with low storage and computational overheads make our scheme to be much suitable for practical applications for linear hierarchies.