Bachelor
2021/2022

# Linear Algebra

Type:
Elective course (International Business and Management Studies)

Area of studies:
Management

Delivered by:
Department of Mathematics

When:
1 year, 1, 2 module

Mode of studies:
offline

Open to:
students of one campus

Language:
English

ECTS credits:
6

Contact hours:
60

### Course Syllabus

#### Abstract

The discipline is aimed at first-year students of the direction "International Business and Management/Management". The purpose of the course is to study basic concepts, master the basic methods of linear algebra and their application for the construction of economic and mathematical models. The course examines the theory of matrices and determinants, the structure of solutions of systems of linear algebraic equations, linear spaces, the basic concepts of analytical geometry on the plane and in space, second-order curves. As a result of learning the discipline, the student have to know the basic concepts of linear algebra, methods for solving systems of linear algebraic equations, elements of vector analysis and analytical geometry; be able to apply the apparatus of linear algebra in problems of forming economic models and solving applied problems; have skills in solving systems of linear equations and constructing diagonal quadratic forms, in applying linear algebra methods in other academic disciplines and scientific work.

#### Learning Objectives

- The goals of learning the discipline "Linear Algebra" are the study of sections of matrix algebra, the solution of systems of linear equations and vector analysis, which allow the student to navigate in such disciplines as" Probability Theory and Mathematical Statistics", "Game Theory and Decision Making". The course "Linear Algebra" will be used in the theory and applications of multidimensional mathematical analysis, differential equations, econometrics. The course materials can be used for the development and application of numerical methods for solving problems from many fields of knowledge, for the construction and research of mathematical models of such problems. The discipline is a model applied apparatus for students of the direction "International Business and Management/Management" to study the mathematical component of their professional education. As learning outcomes the student should: • know the basic concepts of linear algebra, methods for solving systems of linear algebraic equations, elements of vector analysis and analytical geometry; • be able to apply the apparatus of linear algebra in the problems of forming economic models and solving applied problems; • have skills in solving systems of linear equations and constructing diagonal quadratic forms, in applying linear algebra methods in other academic disciplines and scientific work.

#### Expected Learning Outcomes

- Analyzes the sign-definiteness of a quadratic form using the Sylvester criterion
- Applies elementary transformations of systems of linear equations, knows the simplest properties of polynomials, linear operations with them.
- Decomposes polynomials into multipliers, builds an interpolation Lagrange polynomial, selects the integer part of an incorrect rational fraction, decomposes the correct rational fraction into the sum of the simplest fractions by different methods, performs the simplest arithmetic operations with complex numbers, converts a complex number to a trigonometric notation, extracts roots from complex numbers and applies the Moivre formula.
- Determines the orthogonality of vectors; builds an orthogonal projection of a vector onto a subspace; performs the orthogonalization process for a linearly independent system of vectors; builds an orthonormal basis; applies the Gram criterion
- Finds the kernel and image of a linear operator, the eigenvectors and eigenvalues of a linear transformation, writes the matrix of a linear operator with respect to a fixed basis
- Knows the canonical equations and properties of second-order curves; determines the type of second-order curve
- Performs linear operations with vectors in a linear space, analyzes the linear independence of vectors, selects the basis of a finite-dimensional linear space, decomposes the vector according to the basis.
- Performs operations with matrices, performs elementary transformations of matrices, brings the matrix to a stepwise form, finds the rank. Finds the determinant of a square matrix, uses the properties of the determinants, finds the inverse matrix.
- Performs operations with vectors in cartesian coordinate system, uses the properties of operations, finds scalar, vector, scalar triple products of vectors, knows the properties of these products, applies them to solve geometric problems.
- Reduces the quadratic form to a canonical form by means of an orthogonal transformation
- Solves systems of linear algebraic equations by the Gauss method, finds the fundamental system of solutions of homogeneous system of linear algebraic equations, writes down the general solution of homogeneous and inhomogeneous system of linear algebraic equations, applies Kramer's theorem.
- Writes down and uses equations of straight lines and planes, studies their mutual arrangement, builds a set of solutions to systems of linear inequalities on the plane, solves geometry problems using equations of straight lines and planes.

#### Course Contents

- Introduction
- Linear Spaces
- Matrices and determinants
- Systems of Linear Algebraic Equations
- Affine Coordinate Systems.
- Straight lines and Planes
- Polynomials and rational fractions
- Linear Operators
- Quadratic forms.
- Second-order Curves
- Euclidean spaces
- Self-adjoint operators

#### Assessment Elements

- Test 1The test is carried out in the classroom, in writing, 80 minutes. In the case of a distance learning format, control work is carried out remotely. The student must demonstrate the ability to work with matrices and determinants, be able to find a solution to a system of linear equations using Cramer's formulas, find the inverse matrix (through algebraic additions and the method of elementary transformations), and solve matrix equations, find the basis of the vector system and decompose the vector by the basis.
- Test 2The test is carried out in the classroom, in writing, 80 minutes. In the case of a distance learning format, control work is carried out remotely. The student must demonstrate the ability to work with matrices and determinants, be able to find a solution to a system of linear equations using Cramer's formulas, find the inverse matrix (through algebraic additions and the method of elementary transformations), and solve matrix equations, find the basis of the vector system and decompose the vector by the basis.
- Self workThe teacher of the practical seminars evaluates the independent work of students: the completion of homework and preparation for seminars are evaluated. The control can be carried out in the form of oral and written surveys on the material of homework. The accumulated score on a 10-point scale for independent work is determined before the final control – Оself work
- ExamThe exam is conducted in a classroom, in writing, 80 minutes. In the case of a distance learning format, the exam is conducted remotely. On the exam, the student must show knowledge of the theoretical part of the course: knowledge of the formulations of theorems, properties, definitions, basic proofs, be able to apply properties and theorems in practice, be able to solve problems.

#### 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. Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.b.spr.sptbec.978.3.642.20570.5
- Бурмистрова Е. Б., Лобанов С. Г. - ЛИНЕЙНАЯ АЛГЕБРА. Учебник и практикум для академического бакалавриата - М.:Издательство Юрайт - 2019 - 421с. - ISBN: 978-5-9916-3588-2 - Текст электронный // ЭБС ЮРАЙТ - URL: https://urait.ru/book/lineynaya-algebra-425852
- 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