This online course introduces basic concepts and problems in linear algebra: linear space, basis and dimensionality, linear operators, matrices, solving systems of linear equations, constructing a Jordanian normal form, and the study of quadratic forms. Related questions relating to topology and dynamical systems are also considered: the principle of compressive mappings, the study of differential equations, compactness, and Brauer’s theorem. The course is primarily aimed at students beginning to study these topics or who are familiar with them superficially and wish to gain a deeper understanding. In contrast to classical courses in higher mathematics, the lecturer does not strive for rigorous formal presentation of material and systematic coverage of all topics. The presentation is structured around a series of mathematical plots, which are discussed first informally and by example, and only then using strict formulations.
The course includes 16 weeks of lecture and seminar classes. Lectures present basic ideas, examples, plots, and theory. Seminars present more technical issues and problems to illustrate the use of methods. At the end of each week students are offered control tasks, and at the end of all weeks there is a control examination. Tasks are mainly aimed at checking comprehension of materials from lectures and seminars, but there are also more complicated tasks, requiring independent work.
Course program
- Polynomials and linear algebra. Interpolation Lagrange polynomial. Bases and dimensionality of polynomial space. The multiplicity of solutions to a system of linear equations. The dimensionality of linear space.
- Linear operators: definition and assignment by matrix. Composition of linear operators. Exponent from a linear operator. Norm of linear operator and convergence of exponent series.
- Multivariate analysis and linear algebra. Examples: the heat conduction problem, the pendulum problem. Linearization of systems of differential equations.
- Matrices and systems of linear equations. The multiplication and inversion of matrices. Irreducibility and determinant. Algebraic complements and computation of inverse matrices. Matrix of a linear operator in a new basis. Application: cubic interpolation splines.
- Anatomy of a linear operator: diagonalization and Jordanian normal form. Exponent from a matrix and linear dynamical systems.
- Quadratic forms and their matrix notation. Bringing a quadratic form to the canonical form. Rank of a quadratic form. Law of inertia of quadratic forms. Sylvester’s criterion. Reduction to canonical form by orthogonal transformation.
- Metric spaces. Principle of compressive mappings. Its application to the theory of differential equations (proof of existence and uniqueness of solutions).
- Compactness on a straight line and in multidimensional space. The continuous image of a compact.
- Vector fields and their applications: the basic algebra theorem and Brauer’s theorem.