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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

• 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
• 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
• 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 theorems correctly; finds a fundamental solution system; analyzes the structure of the solution set of the system of linear equations
• 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
• 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
Matrix and extended matrix of a system of linear equations. Operations with matrices (sum of matrices, multiplication of a matrix by a number, product of matrices). Properties of arithmetic operations on matrices. Matrix notation of the system of equations.
• Determinant
Permutations. Inversion, parity of permutation. Determinant and elementary transformations. Determinant computation by column decomposition. Determinant of the transposed matrix. Determinant computation by string decomposition. Inverse and mutual matrices. Cramer's formulas. Construction of the inverse matrix by elementary transformations. Determinant of the product of matrices.
• Linear Spaces
Definition of linear space, examples. The simplest consequences of the axioms of a linear space. Subspace of linear space. Linear dependence and independence of vectors. Properties. Linear span. Basis and dimension of linear space. Existence of a basis for a finite-dimensional space.
• Matrix Rank
The rank of the matrix. Elementary matrix transformations. Reversibility of elementary transformations. Transforming of matrices to a stepped form by elementary transformations. Consistency of rank under elementary transformations. The rank of the stepped matrix. The theorem of matrix rank. A criterion for linear independence of a system of rows (columns). The rank of the product of matrices.
• Systems of Linear Algebraic Equations
Various forms of notation of systems of linear algebraic equations. The Kronecker-Capelli theorem on the compatibility of a system of linear equations. Dimension and structure of the space of solutions of a homogeneous system of linear equations. Gauss method for solving systems of linear equations. The structure of the general solution of an inhomogeneous system of linear equations.
• Euclidean Spaces
Scalar product. Examples and properties of Euclidean spaces. Cauchy-Bunyakovsky inequality. Triangle inequality. Vector length and angle between vectors. Orthogonality of vectors. Independence of pairwise orthogonal vectors. Orthogonal projection of a vector onto a subspace. Orthogonalization of the basis. Orthogonal matrices and their properties. Geometric interpretation of orthogonal matrices.
• Linear Operators
Linear operators and linear transformation. Examples of linear operators. Kernel and image of a linear operator, their properties. Linear transformation matrix. Transformation of the matrix of a linear operator in the changed the basis. Composition matrix of linear operators. Linear functionals. Transformation of the matrix of a linear operator to a diagonal form. Eigenvectors and eigenvalues of a linear operator. Characteristic polynomial of a linear operator. About the roots of the characteristic polynomial of a linear operator. Perron's theorem.
Quadratic forms. A symmetric matrix generating a quadratic form. The concept of sign-definite quadratic forms. Sylvester's criterion. Application of quadratic forms to the functions analysis.
Conjugacy of operators in Euclidean space. Transformation of a symmetric matrix to a diagonal form. Symmetric matrices and self-adjoint operators in Euclidean space. Orthonormal basis of eigenvectors of a self-adjoint operator. Convertation of the quadratic form to the canonical form.
• Second-order Curves
Geometric interpretation of the solution set of a second-order algebraic equation with two unknown variables. Ellipse. Circle. Hyperbola. Parabola. Determination of the line type for a given algebraic equation of the second order. The concept of second-order surfaces.
• Jordan Form of a Matrix
Jordan block. Adjoint vectors. Canonical Jordan form of a matrix. Algorithm for calculating the dimension of cells and constructing an orthogonal transformation. Matrix functions. Examples and applications.
• Linear Operator Norm
The norm of a linear operator. Its simplest properties (norm of a linear combination of an operator, norm of a composition of an operator). Estimation of the norm of an operator using elements of its matrix. Self-adjoint operator norm. Examples of calculating the 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

• Interim assessment (2 module)
0.46 * Exam + 0.14 * Self Work + 0.1 * Test 1 + 0.15 * Test 2 + 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