Linear Algebra

2024/2025

Academic Year

ENG

Instruction in English

3

ECTS credits

Delivered at:

International College of Economics and Finance

Course type:

Compulsory course

When:

2 year, 1, 2 module

Instructors

Брыков Вячеслав Вячеславович

Pervouchine, Dmitry D.

Ramazyan, Tigran

Course Syllabus

Abstract

Pre-requisites There are no prerequisite courses for Linear Algebra. Nonetheless, some concepts of Calculus and Statistics will be used as illustrations. Therefore, Linear Algebra is recommended for the audience that are familiar with these disciplines. Course description Linear Algebra is a half-semester (12 weeks) class that is obligatory for the curriculum of the second-year ICEF students. The course was originally designed as an instrumental supplement to the principal quantitative block subjects such as “Methods of optimization”, “Time series analysis”, and “Econometrics”. The class of Linear Algebra in ICEF is taught on its own to deliver basic principles of matrix calculus. From a broader perspective, the aim of the course is to deliver one of the most general mathematical concepts - the idea of linearity. The course splits naturally into the following three parts: 1. Problems related to systems of linear equations and to the extension of the 2D- and 3D- intuition to linear spaces of higher dimensions. This part includes the concepts of basis, rank, dimension, linear hull, linear subspace, etc. 2. Problems that involve antisymmetric polylinear forms (determinants) and also problems from the geometry of linear operators such as eigenvectors and eigenvalues, matrix diagonalization, etc. 3. Problems from the calculus of bilinear forms: quadratic forms, orthogonalization, and other geometric problems in higher-dimensional Euclidean spaces.

Learning Objectives

• Students are expected to develop an understanding of basic algebraic concepts such as linear vector space, linear independence, bases, coordinate systems, dimension, matrix algebra, linear operators, dot product, orthogonality. On the practical side, among other skills, students are expected to be able to solve systems of linear equations, find fundamental systems of solutions, invert matrices, find eigenvalues, and do orthogonal projections.

Expected Learning Outcomes

• Be able to test linear independence
• Compute matrix determinants and inverse matrices
• Do orthogonal projections and find orthogonal bases
• Find eigenvalues and diagonalize matrices
• Solve systems of linear equations

Course Contents

• Systems of linear equations in matrix form.
• Linear space. Linear independence.
• Linear subspace
• Matrix as a set of columns and as a set of rows.
• Determinant of a set of vectors.
• Inverse matrix
• Linear operator as a geometric object.
• Eigenvalues, eigenvectors and their properties
• Bilinear and quadratic forms.
• Dot product in linear spaces.

Assessment Elements

• Mock exam
• Home Assignments
• Final Exam
  In order to get a passing grade for the course, the student must sit (all parts) of the examination.

Interim Assessment

• 2024/2025 2nd module
  0.5 * Final Exam + 0.1 * Home Assignments + 0.4 * Mock exam

Bibliography

Recommended Core Bibliography

• Mathematics for economists, Simon, C. P., 1994

Recommended Additional Bibliography

• Лекции по линейной алгебре, Гельфанд, И. М., 1998

Authors

• PERVUSHIN DMITRIY DAVIDOVICH