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# Linear Algebra

2021/2022
Учебный год
ENG
Обучение ведется на английском языке
6
Кредиты
Статус:
Курс по выбору
Когда читается:
1-й курс, 1, 2 модуль

### Course Syllabus

#### Abstract

The course is focused on first-year students of the Economics department. 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 "Economics" department students.

#### Expected Learning Outcomes

• Analyzes the sign-definiteness of a quadratic form using Sylvester's criterion
• 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
• Converts a matrix to Jordan form
• Converts a quadratic form to a canonical form using an orthogonal transformation
• 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
• Estimates the operator norm
• 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
• Knows canonical equations and properties of second-order curves; defines the type of the second-order curve
• Performs elementary matrix transformations correctly and efficiently; transforms the matrix to a stepped form; determines the matrix rank in different ways
• Performs operations with matrices; finds matrices with given properties
• Solves system of linear equations applying Gauss method; uses the Kronecker-Capelli theorems correctly; finds a fundamental solution system; analyzes the structure of the solution set of the system of linear equations

#### Course Contents

• Matrices and Systems of Linear Equations
• Determinant
• Linear Spaces
• Matrix Rank
• Systems of Linear Algebraic Equations
• Euclidean Spaces
• Linear 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 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 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.
• Self Work
The teacher of the practical seminars evaluates the independent 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 10 points (the arithmetic mean for the surveys is set, rounding is from 0.6). 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, 80 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.

#### Interim Assessment

• 2021/2022 2nd module
0.46 * Exam + 0.1 * Test 1 + 0.15 * Test 3 + 0.15 * Test 2

#### 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