This course is an introduction to some basic facts about real functions, real analysis and linear algebra with applications to finding solutions to linear systems.
The course is devoted to provide a basic knowledge of the main mathematical tools with specific attention to the concept of mathematical function, of limit, and of differential and integral calculus. Part of the course gives the students the main techniques to solve linear systems in many variables.
The course starts with a short recall of some basic facts about set theory (union, intersection, function, injective and surjective function, composition of functions and invertible function). Then we recall some properties of rational and real numbers. The first part of the course deals with real functions. We define the notion of limit, continuity, differentiability and regularity. We will show that Taylor's theorem provides a very useful tool to compute limits of a fairly regular function. Then we show the classical results concerning the study of the graph of a function (extremal points, asymptotes, convexity). Finally, we will explain the theory of integration according to Riemann and some techniques to extract the primitive function of a given one. In the second part of the course we will explain some basic notions in linear algebra with particular emphasis on the resolution of linear systems. We will introduce the notion of vector space, linear map, rank, kernel and determinant.