The book is devoted to the presentation of fundamental concepts and apparatus of linear algebra and related sections of geometry. The book differs from the existing courses of linear algebra by great attention to applications and connections with other areas of mathematics: a discussion of the basic principles of quantum mechanics is included, the geometry of Minkowski space is described, and an introduction to linear programming is given. The book contains modern mathematical material that was not presented in traditional manuals: category language and categorical properties of linear spaces, Kähler metric, an introduction to Hilbert polynomial theory.