Algebra and Calculus
Objectives
The aim of this course is to provide students with additional algebraic and analytical skills that will be useful for following courses in probability, statistics and optimization. At the end of the course, students should be able to use elementary techniques for reducing endomorphisms, as well as the basic properties of orthogonal projectors, which will be used 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 the integration and probability courses. Basic notions 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, symmetric positive-definite matrices, definition of a Euclidean space, scalar product, norm, orthogonality, orthogonal and orthonormal bases.
3. Projections: definition, properties in terms of rank, similar matrices, properties of matrices of these applications on a normed vector space, characteristic in terms of norm, orthogonal projection theorem, application to simple linear regression.
4. Numerical series: absolute convergence, series/integral comparison.
5. Function sequences: simple and uniform convergence, transmission of continuity and derivation, integral and limit inversion on a bounded interval.
6. Integral series: radius of convergence, usual integral series developments.
7. Continuity and derivability of functions in several variables: partial derivatives, Ck class function, Jacobian matrix, limited development.
Prerequisites
The algebra and analysis part of the competitive examination curriculum, or the concepts of refresher courses in mathematics.