Electrical circuits, control loops, vibrating structures, and periodic signals all share the same mathematical basis: linear systems, seen in time and frequency. This book unites linear algebra, ordinary differential equations, and Fourier theory within a single framework for engineers. Rather than treating these topics separately, the text tells a single story driven by engineering questions. Is this equilibrium stable? Which frequencies pass through a system? Each concept appears exactly when it becomes necessary, starting from the equation y' = Ay + g in the opening pages. Designed for upper-level undergraduate and Master's students in engineering, the book closes the gap between foundational mathematics and real systems, equipping readers not only to carry out calculations but also to interpret their results.