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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 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

• Applies elementary transformations of systems of linear equations, knows the simplest properties of polynomials, linear operations with them.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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
• Analyzes the sign-definiteness of a quadratic form using the Sylvester criterion
• Knows the canonical equations and properties of second-order curves; determines the type of second-order curve
• 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
• Reduces the quadratic form to a canonical form by means of an orthogonal transformation

#### Course Contents

• Introduction
Motivational examples (application of LA and geometry methods in professional activity). Elementary transformations of systems of linear equations. An idea of the Gauss method. The simplest properties of polynomials, linear operations with them.
• Linear Spaces
Free geometric vectors. Linear operations on vectors. The concept of linear space and subspaces, examples. Linear dependence and independence of vector systems. Properties. The basis and rank of the vector system. Properties of the basis.
• Matrices and determinants
Matrices and linear operations on them. Matrix multiplication, properties. Determinants of square matrices: properties, calculation methods. The inverse matrix: the criterion of existence and methods of construction. The rank of the matrix. Definitions and calculation methods. Elementary transformations of matrices. Reduction of the matrix to a trapezoidal shape (stepped form).
• Systems of Linear Algebraic Equations
Various forms of notation of systems of linear algebraic equations. Conditions of compatibility and certainty. The Gauss method for solving systems of linear equations. The structure of the general solution of a homogeneous system of linear equations. The structure of the general solution of an inhomogeneous system of linear equations. Square systems of linear equations. Kramer's theorem. An idea of the least squares method. An idea of the Leontiev model of a diversified economy.
• Affine Coordinate Systems.
The relations between a vector and a point (affine) space. Affine coordinate systems. The relations between the coordinates of a point in different affine coordinate systems. Cartesian rectangular coordinate system. Scalar product of geometric vectors, its properties and application. Representation of the vector and scalar triple product of vectors. Properties and application. Setting lines and surfaces using equations. Examples.
• Straight lines and Planes
Straight lines on the plane. Various forms of equations. The relative position of straight lines on the plane. The distance from the point to the straight line. The geometric meaning of inequalities of the first degree. Systems of linear inequalities. Planes in space. Various forms of equations. The relative position of the planes. The distance from the point to the plane. Straight lines in space. Parametric and canonical equations of a straight line. The relative position of straight lines in space. A segment in space. A hyperplane in Rn. Mutual arrangement of hyperplanes. A straight line in Rn. The mutual arrangement of straight lines. The segment in Rn. Convex sets. The concept of a convex polyhedron. Presentation of the linear programming problem, solution methods, sensitivity analysis.
• Polynomials and rational fractions
The binomial of Newton. Properties of polynomials (relative to a real variable). Factorization of polynomials. The interpolation problem. The Lagrange interpolation polynomial. Rational fractions and their properties. Selection of the integer part of an incorrect rational fraction. Decomposition of a regular rational fraction into the simplest ones. The method of undefined coefficients and the method of strikethrough. Definition of complex numbers and their properties. Geometric representation of complex numbers. The trigonometric form of the record. The product and quotient theorem. The Moivre formula. Extracting the root of a complex number. Properties of polynomials (with respect to a complex variable).
• Linear Operators
Linear operator and linear transformation. Examples of linear operators. The kernel and the image of a linear operator, their properties. A linear transformation matrix. Eigenvectors and eigenvalues of the linear transformation. The characteristic polynomial. Properties of eigenvectors. The idea of a linear model of international trade. Transformation of the linear transformation matrix when replacing the basis. The matrix of the composition of linear operators. Reduction of the linear transformation matrix to a diagonal form.
Quadratic forms. A symmetric matrix that generates a quadratic form. Representation of sign-defined quadratic forms. The Sylvester criterion. Application of quadratic forms for the analysis of functions.
• Second-order Curves
Geometric interpretation of the set of solutions of an algebraic equation of the second order with respect to two unknowns. An ellipse. The circle. Hyperbole. A parabola. Determination of the line type by a given second-order algebraic equation. Representation of second-order surfaces.
• Euclidean spaces
Examples and properties of Euclidean spaces. The Cauchy-Bunyakovsky inequality. The triangle inequality. Independence of pairwise orthogonal vectors. Orthogonal projection of a vector onto a space. Orthogonalization of the basis. Geometric interpretation of orthogonal matrices
Conjugacy of operators in Euclidean space. Eigenvectors of self-adjoint operators. Reduction of a quadratic form to a canonical form.

#### 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. 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 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. 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 work
The 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
• 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. 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

• Interim assessment (2 module)
0.4 * Exam + 0.2 * Self work + 0.2 * Test 1 + 0.2 * 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