Ordinary differential equations (ODEs) and linear algebra are taught in foundational post-calculus mathematics courses in the sciences. This text will help students master both subject areas. Linear algebra is systematically developed first, with an eye towards solving linear systems of ODEs, and over fifteen distinct applications of the theory are explored, including lead poisoning, SIR models and digital filters, not typically seen in textbooks at this level. A large emphasis is also placed on mathematical modeling, with the group projects at the end of each chapter allowing students to fully explore the interaction between the modeling of a system, the solution of the model, and the resulting physical description. The seamless transition between linear algebra and ODEs contained within this one text makes this a valuable resource for students who have had one year of calculus and are taking a first class in ODEs.