Humans and animals engage in rich social interactions. It is often theorized that a relatively small number of basic social interactions give rise to the full range of behavior observed. But no computational theory explaining how social interactions combine together has been proposed before. We do so here. We take a model, the Social MDP, which is able to express a range of social interactions, and extend it to represent linear combinations of social interactions. Practically for robotics applications, such models are now able to not just express that an agent should help another agent, but to express goal-centric social interactions. Perhaps an agent is helping someone get dressed, but preventing them from falling, and is happy to exchange stories in the meantime. How an agent responds socially, should depend on what it thinks the other agent is doing at that point in time. To encode this notion, we take linear combinations of social interactions as defined in Social MDPs, and compute the weights on those combinations on the fly depending on the estimated goals of other agents. This new model, the Linear Social MDP, enables zero-shot reasoning about complex social interactions, provides a mathematical basis for the long-standing intuition that social interactions should compose, and leads to interesting new behaviors that we validate using human observers. Complex social interactions are part of the future of intelligent agents, and having principled mathematical models built on a foundation like MDPs will make it possible to bring social interactions to every robotic application.