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Bachelor 2022/2023

# Linear Algebra

Category 'Best Course for New Knowledge and Skills'
Area of studies: Economics
When: 1 year, 1, 2 module
Mode of studies: offline
Open to: students of one campus
Instructors: Леонова Екатерина Олеговна, Кузнецова Мария Станиславовна, Леонова Екатерина Олеговна, Alexandra Grinikh, Кумачева Сурия Шакировна
Language: English
ECTS credits: 6
Contact hours: 112

### Course 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
The 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. In the latter case, it can consist of two parts - written (extracurricular) and oral (classroom, including online). 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. Test is carried out in writing form. 80 minutes are supposed for writing test and 10 minutes are supposed for loading the solved tasks. The work is carried out on the platform Smart LMS and Zoom or MS Teams. Students should connect to the lesson in which the test will be carried out 15 minutes before the start, at the lector's signal, start solving tasks in Smart LMS. The student's computer must meet the requirements: the presence of a working camera and microphone, high-speed Internet, support for Zoom / MS Teams. Headphones cannot be used. The answers to the tasks are written on white A4 sheets, with a black pen, the sheets are numbered, when indicated in the task, the answers are additionally entered into the answer window. After finishing the work, the student must take a photo / scan of his/her solution and upload it to Smart LMS. Photos should be vertical so that the text is not blurry and unambiguous to read. The answers and task numbers must be highlighted. During operation, the camera and microphone must be turned on. It is required to position the camera to the side or in front of you in such a way that it is directed to the working field - the sheet on which the work is performed, to the student and the space around the student (the room must be well lit). There must be one camera! (It is allowed to use the login to Zoom / MS Teams from a mobile phone with its camera, if there is no webcam at the computer). At the request of the lector, the student is obliged to switch to broadcasting his screen: turn on the back camera of the mobile phone or turn the phone to the computer screen within 5 seconds, or start a screen sharing. Student cannot leave the room during the test. Student can have only blank sheets of paper, water and writing instruments on the table without pencil case. The presence of any media in the vicinity of the student's workplace, as well as other people, is considered a violation and leads to the removal of the student from the test and giving a grade of "0". During the test, students are prohibited from turning off the camera and microphone: until the end of the work, video and sound must remain active, including the time for scanning the completed job and submit it for review. A short-term communication interruption during the test is considered to be a communication interruption of less than 5 minutes and no more than once. A long-term interruption of communication during the test is considered to be a violation of 5 minutes or more. In case of a long-term disruption of communication, the student can continue to participate in the writing of the work only at the decision of the teacher.
• Test 2
The 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. In the latter case, it can consist of two parts - written (extracurricular) and oral (classroom, including online). The student must be able to calculate the matrix rank, find the general solution of a system of equations as the sum of a particular solution and a fundamental system of solutions, apply the Gauss method for solving a system of linear equations, investigate a system of vectors for linear dependence, find expansions in a basis in a linear space, find a transformation matrix from one basis in linear space to another, find the eigenvalues of the matrix. Test is carried out in writing. 80 minutes are supposed for writing test and 10 minutes are supposed for loading the solved tasks. The work is carried out on the platform Smart LMS and Zoom or MS Teams. Students should connect to the lesson in which the test will be carried out 15 minutes before the start, at the lector's signal, start solving tasks in Smart LMS. The student's computer must meet the requirements: the presence of a working camera and microphone, high-speed Internet, support for Zoom / MS Teams. Headphones cannot be used. The answers to the tasks are written on white A4 sheets, with a black pen, the sheets are numbered, when indicated in the task, the answers are additionally entered into the answer window. After finishing the work, the student must take a photo / scan of his/her solution and upload it to Smart LMS. Photos should be vertical so that the text is not blurry and unambiguous to read. The answers and task numbers must be highlighted. During operation, the camera and microphone must be turned on. It is required to position the camera to the side or in front of you in such a way that it is directed to the working field - the sheet on which the work is performed, to the student and the space around the student (the room must be well lit). There must be one camera! (It is allowed to use the login to Zoom / MS Teams from a mobile phone with its camera, if there is no webcam at the computer). At the request of the lector, the student is obliged to switch to broadcasting his screen: turn on the back camera of the mobile phone or turn the phone to the computer screen within 5 seconds, or start a screen sharing. Student cannot leave the room during the test. Student can have only blank sheets of paper, water and writing instruments on the table without pencil case. The presence of any media in the vicinity of the student's workplace, as well as other people, is considered a violation and leads to the removal of the student from the test and giving a grade of "0". During the test, students are prohibited from turning off the camera and microphone: until the end of the work, video and sound must remain active, including the time for scanning the completed job and submit it for review. A short-term communication interruption during the test is considered to be a communication interruption of less than 5 minutes and no more than once. A long-term interruption of communication during the test is considered to be a violation of 5 minutes or more. In case of a long-term disruption of communication, the student can continue to participate in the writing of the work only at the decision of the teacher.
• Test 3
The 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. In the latter case, it can consist of two parts - written (extracurricular) and oral (classroom, including online). The student should demonstrate the ability to apply the Gram-Schmidt orthogonalization method, find the projection of a vector onto a subspace, find the eigenvalues of an operator, find the matrix of a linear operator in the indicated bases, transform a quadratic form to a diagonal form by orthogonal transformation, and investigate it for definiteness. Test is carried out in writing. 80 minutes are supposed for writing test and 10 minutes are supposed for loading the solved tasks. The work is carried out on the platform Smart LMS and Zoom or MS Teams. Students should connect to the lesson in which the test will be carried out 15 minutes before the start, at the lector's signal, start solving tasks in Smart LMS. The student's computer must meet the requirements: the presence of a working camera and microphone, high-speed Internet, support for Zoom / MS Teams. Headphones cannot be used. The answers to the tasks are written on white A4 sheets, with a black pen, the sheets are numbered, when indicated in the task, the answers are additionally entered into the answer window. After finishing the work, the student must take a photo / scan of his/her solution and upload it to Smart LMS. Photos should be vertical so that the text is not blurry and unambiguous to read. The answers and task numbers must be highlighted. During operation, the camera and microphone must be turned on. It is required to position the camera to the side or in front of you in such a way that it is directed to the working field - the sheet on which the work is performed, to the student and the space around the student (the room must be well lit). There must be one camera! (It is allowed to use the login to Zoom / MS Teams from a mobile phone with its camera, if there is no webcam at the computer). At the request of the lector, the student is obliged to switch to broadcasting his screen: turn on the back camera of the mobile phone or turn the phone to the computer screen within 5 seconds, or start a screen sharing. Student cannot leave the room during the test. Student can have only blank sheets of paper, water and writing instruments on the table without pencil case. The presence of any media in the vicinity of the student's workplace, as well as other people, is considered a violation and leads to the removal of the student from the test and giving a grade of "0". During the test, students are prohibited from turning off the camera and microphone: until the end of the work, video and sound must remain active, including the time for scanning the completed job and submit it for review. A short-term communication interruption during the test is considered to be a communication interruption of less than 5 minutes and no more than once. A long-term interruption of communication during the test is considered to be a violation of 5 minutes or more. In case of a long-term disruption of communication, the student can continue to participate in the writing of the work only at the decision of the teacher.
• Self Work
The teacher of the practical seminars evaluates the self work of students: the performance of homework and preparation for the seminars is assessed. The control can be carried out in the form of oral and written tests on the material of homework, as a result of which the student can accumulate 8 points (the arithmetic mean for the surveys is set, rounding is from 0.6). The student gains the remaining two points (up to ten points) by completing additional tasks from the homework (as directed by the teacher). The arithmetic mean of the polls is set, rounding is from 0.6. For grading assignments, the general grading criteria given above is applied. The accumulated score on a 10-point scale for self work is determined before the final control – Оself work
• Exam
The exam is conducted in a classroom, in writing, 90 minutes. In the case of a distance learning format, the exam is conducted remotely. In the latter case, it can consist of two parts - written (extracurricular) and oral (classroom, including online). 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. Test is carried out in writing. 80 minutes are supposed for writing test and 10 minutes are supposed for loading the solved tasks. The work is carried out on the platform Smart LMS and Zoom or MS Teams. Students should connect to the lesson in which the test will be carried out 15 minutes before the start, at the lector's signal, start solving tasks in Smart LMS. The student's computer must meet the requirements: the presence of a working camera and microphone, high-speed Internet, support for Zoom / MS Teams. Headphones cannot be used. The answers to the tasks are written on white A4 sheets, with a black pen, the sheets are numbered, when indicated in the task, the answers are additionally entered into the answer window. After finishing the work, the student must take a photo / scan of his/her solution and upload it to Smart LMS as a PDF-file. Photos should be vertical so that the text is not blurry and unambiguous to read. The answers and task numbers must be highlighted. During operation, the camera and microphone must be turned on. It is required to position the camera to the side or in front of you in such a way that it is directed to the working field - the sheet on which the work is performed, to the student and the space around the student (the room must be well lit). There must be one camera! (It is allowed to use the login to Zoom / MS Teams from a mobile phone with its camera, if there is no webcam at the computer). At the request of the lector, the student is obliged to switch to broadcasting his screen: turn on the back camera of the mobile phone or turn the phone to the computer screen within 5 seconds, or start a screen sharing. Student cannot leave the room during the test. Student can have only blank sheets of paper, water and writing instruments on the table without pencil case. The presence of any media in the vicinity of the student's workplace, as well as other people, is considered a violation and leads to the removal of the student from the test and giving a grade of "0". During the test, students are prohibited from turning off the camera and microphone: until the end of the work, video and sound must remain active, including the time for scanning the completed job and submit it for review. A short-term communication interruption during the test is considered to be a communication interruption of less than 5 minutes and no more than once. A long-term interruption of communication during the test is considered to be a violation of 5 minutes or more. In case of a long-term disruption of communication, the student can continue to participate in the writing of the work only at the decision of the teacher.

#### Interim Assessment

• 2022/2023 2nd module
0.15 * Test 2 + 0.14 * Self Work + 0.46 * Exam + 0.1 * Test 1 + 0.15 * Test 3

#### 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