This chapter examines linearly coupled fluid-structure interaction (FSI) models, where the interaction between fluid and structure is approximated using linear theories, simplifying analysis while retaining key physical insights. The fluid is modeled on a fixed domain, avoiding the complexities of moving boundaries. These models are useful in scenarios with small deformations (e.g., blood flow in certain scenarios, wind effects on some structures) but still pose challenges in well-posedness due to two-way coupling and multiphysics interactions. A prototypical linearly coupled FSI problem is introduced, featuring a lower-dimensional elastic structure on the boundary of the fluid domain. The Lie operator splitting method is introduced here to construct weak solutions for the linear case, laying the groundwork for later analysis of more complex models.