This course develops the basic geometric, algebraic, and computational foundations of vector spaces and matrices and applies them to a wide range of problems and models. The material will be accessible to students who have taken at least one semester of calculus and is useful to most consumers of mathematics. The course focuses on real finite dimensional vector spaces and inner product spaces, although abstract and infinite-dimensional vector spaces will be discussed towards the end of the semester. Applications will be made to computer graphics, environmental models, differential equations, Fourier series, and physics. Computers will be used throughout. Problem sets will be assigned for almost every class. Prerequisite: a year of Calculus.