This paper presents a new achievable scheme for the K-user Linear Computation Broadcast Channel (K-LCBC). A K-LCBC comprises data stored on a server and K users, each aiming to retrieve a desired linear function of the data by leveraging their prior locally available side information in the form of another linear function of the data. The proposed scheme is based on a subspace decomposition derived from representable polymatroid spaces. This decomposition enables the server to effectively design multicast messages that simultaneously benefit multiple users and allow users to eliminate interference using their available side information. This work extends existing results for the 3-LCBC by introducing a linear programming framework to optimize multicast opportunities across an arbitrary number of users. The proposed approach can be used to derive achievable scheme for the K-user coded caching problem with linear coded placement and scalar linear function retrieval, which was our original motivation to investigate the K-LCBC.