Algebra and Calculus

Objectives

This course aims to provide useful algebra and analysis supplements for following courses in probability, statistics and optimization. At the end of this course, students should know how to use elementary techniques for reducing endomorphisms as well as the basic properties of orthogonal projectors, which will be covered in multivariate exploratory statistics. For the analysis part, students will be able to study simple cases of convergence of sequences or series of functions, essential for courses in integration and probability. The basic concepts concerning the derivation of functions of several variables will also be covered and will be fundamental for courses in numerical methods, inferential statistics and all second-year regression courses.

Course outline

1. Reduction of endomorphisms: eigenvalues, eigensubspaces, diagonalizability criterion, characteristic polynomial, similar matrices, matrix polynomials. 2. Scalar product and orthogonality: bilinear and quadratic forms, positive definite symmetric matrices, definition of a Euclidean space, of the scalar product, norm, orthogonality, orthogonal and orthonormal bases. 3. Projections: definition, properties in terms of rank, of similar matrices, properties of the matrices of these applications on a normed vector space, characteristic in terms of norm, theorem of orthogonal projection, application to simple linear regression. 4. Numerical series: absolute convergence, series/integral comparison. 5. Function sequences: simple and uniform convergence, transmission of continuity, of derivation, integral inversion and limit on a bounded interval. 6. Power series: radius of convergence, usual power series developments. 7. Continuity and derivability of functions with several variables: partial derivatives, class function Ck, Jacobian matrix, limited development.

Prerequisites

The algebra and analysis part of the competition program or the concepts of remedial teaching in mathematics