Linearly homomorphic signature (LHS) enables linear computation on signed data, and they have been investigated in many flavors and settings, e.g., for network coding to resist pollution attacks and computation on outsourced data. The security of traditional LHS depends entirely on the assumption that the secret signing key is completely secure. Exposure of signing keys requires updating all signatures that have been generated. However, as relatively insecure mobile devices are increasingly used in network coding and data outsourcing systems, the key exposure issue is becoming more prevalent. To reduce the hazard of key exposure in LHS setting, we integrate key update into LHS, and present forward secure linearly homomorphic signature (FSLHS). Specifically, we first formalize the definition and security notions for the FSLHS scheme, and give a concrete scheme. We then prove our proposed scheme to be forward secure against adaptively chosen message attack in cases where the adversary is able to forge two types of signatures, assuming the CDH hardness problem. Moreover, compared with previous related works, the performance analysis shows that all parameters of our scheme enjoy complexities of log magnitude with respect to the total number of time periods.