This class is one of our core classes, and it serves as an introduction to concepts and tools from system theory. It covers: Advanced linear algebra, including matrix algebra and analysis, and theory of linear operators. The first part of the course is primarily concerned with developing these important tools. It forms the technical bulk of the class, and developing this direction is as much a focus and goal of the course as anything else. Then, we cover basics of linear systems in the time domain, including solution to LTI systems, Lyapunov equations and stability, reachability/observability. Finally, we introduce the basics of optimal control, including the principle of optimality and the HJB equation. We consider the LQR and Riccatti equations in detail. In the process, topics like Semidefinite matrices are explored.