Delivered at:: Department of Mathematics

Course type:: Compulsory course

When:: 1 year, 2, 3 module

Instructors

Grinikh, Alexandra

Кумачева Сурия Шакировна

Full Syllabus

Abstract

The course is focused on first-year students of the Economics and Management of the International Bachelor's in Business and Economics. The goal of the course is to study basic concepts, learning the basic methods of Linear Algebra and their application for the construction of economic and mathematical models. The course studies the theory of matrices and determinants, the structure of solutions to systems of linear algebraic equations, linear spaces, Euclidean spaces, the theory of self-adjoint operators and the second order curves.

Learning Objectives

The objectives of learning the course "Linear Algebra" are the study of units of Matrix Algebra, the solution of systems of Linear Equations and Vector Analysis, allowing the student to navigate in such courses as "Probability Theory and Mathematical Statistics", "Methods of Optimal Solutions", "Mathematical Models in Economics". The course "Linear Algebra" is used in the theory and applications of multivariate mathematical analysis, differential equations, mathematical economics, econometrics. The content of the course can be used to design and apply numerical methods for solving problems from many areas of knowledge, for constructing and researching mathematical models of similar problems. The discipline is a model applied apparatus for studying the mathematical components of professional education of students of the Economics and Management.
1. know the Matrix Equations solving theory, the fundamentals of Vector Analysis and Analytical Geometry
2. be able to apply the linear algebra tools in the forming of economic models and solving of applied problems used in the courses "Mathematical Models in Economics" and "Game Theory"
3. have skills in solving systems of linear equations and constructing diagonal quadratic forms

Expected Learning Outcomes

Performs operations with matrices; finds matrices with given properties
Calculates the determinant; uses the properties of the determinant correctly; solves matrix equations, finds inverse matrix; explores systems of linear algebraic equations applying Cramer's formulas
Checks the linear independence of elements; determines whether the set with the operations introduced on it is a linear space; finds a basis of a finite-dimensional linear space / subspace; decomposes a vector in a basis; builds a transformation matrix when the basis changes
Performs elementary matrix transformations correctly and efficiently; transforms the matrix to a stepped form; determines the matrix rank in different ways
Solves system of linear equations applying Gauss method; uses the Kronecker-Capelli (Rouché–Capelli) theorems correctly; finds a fundamental solution system; analyzes the structure of the solution set of the system of linear equations
Factorizes polynomials, extracts the integer part of an improper rational fraction, decomposes a proper rational fraction into the sum of simple fractions using different methods, performs simple arithmetic operations with complex numbers, converts a complex number to trigonometric notation, extracts roots from complex numbers and applies De Moivre's formula
Determines the orthogonality of vectors; builds an orthogonal projection of a vector onto a subspace; conducts an orthogonalization process for a linearly independent system of vectors; builds an orthonormal basis; applies Gram's criterion
Construct and uses equations of lines and planes, investigates their relative position, builds a set of solutions to systems of linear inequalities on a plane, solves problems of analytic geometry using equations of lines and planes, applies inner product, cross product and triple product
Finds the matrix of a linear operator in a fixed basis and in the transformed basis; finds the kernel and image of a linear operator; finds the eigenvalues and eigenvectors of a linear transformation
Analyzes the sign-definiteness of a quadratic form using Sylvester's criterion
Converts a quadratic form to a canonical form using an orthogonal transformation
Knows canonical equations and properties of second-order curves; defines the type of the second-order curve
Converts a matrix to Jordan form
Estimates the operator norm

Course Contents

Matrices and Systems of Linear Equations
Determinant
Linear Spaces
Matrix Rank
Systems of Linear Algebraic Equations
Polynomials and Rational Fractions
Euclidean Spaces
Analytic Geometry
Linear Operators
Quadratic Forms
Self-adjoint Operators
Second-order Curves
Jordan Form of a Matrix
Linear Operator Norm

Assessment Elements

Test 1
Test 2
Discussions at seminars
Exam

Interim Assessment

2023/2024 3rd module
0.05 * Discussions at seminars + 0.05 * Discussions at seminars + 0.4 * Exam + 0.25 * Test 1 + 0.25 * Test 2

Bibliography

Recommended Core Bibliography

Williams, G. (2019). Linear Algebra with Applications (Vol. Ninth edition). Burlington, MA: Jones & Bartlett Learning. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1708709

Recommended Additional Bibliography

Fuad Aleskerov, Hasan Ersel, & Dmitri Piontkovski. (2011). Linear Algebra for Economists (Vol. 2011). Springer.
Бурмистрова, Е. Б. Линейная алгебра : учебник и практикум для академического бакалавриата / Е. Б. Бурмистрова, С. Г. Лобанов. — Москва : Издательство Юрайт, 2019. — 421 с. — (Бакалавр. Академический курс). — ISBN 978-5-9916-3588-2. — Текст : электронный // Образовательная платформа Юрайт [сайт]. — URL: https://urait.ru/bcode/425852 (дата обращения: 28.08.2023).
Линейная алгебра и аналитическая геометрия. Практикум: Учебное пособие / А.С. Бортаковский, А.В. Пантелеев. - М.: НИЦ ИНФРА-М, 2015. - 352 с.: 60x90 1/16. - (Высшее образование: Бакалавриат). (переплет) ISBN 978-5-16-010206-1 - Режим доступа: http://znanium.com/catalog/product/476097
Основы линейной алгебры и аналитической геометрии: Учебно-методическое пособие / В.Г. Шершнев. - М.: НИЦ ИНФРА-М, 2013. - 168 с.: 60x88 1/16. - (Высшее образование: Бакалавриат). (обложка) ISBN 978-5-16-005479-7 - Режим доступа: http://znanium.com/catalog/product/318084

Bachelor’s Programme 'International Bachelor's in Business and Economics'

Contacts:

Linear Algebra