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# Linear Algebra

2023/2024
Учебный год
ENG
Обучение ведется на английском языке
7
Кредиты
Статус:
Курс обязательный
Когда читается:
1-й курс, 2, 3 модуль

### 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
• 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
0.4 * Exam + 0.05 * Discussions at seminars + 0.25 * Test 1 + 0.25 * 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