The Government of the Russian Federation

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1 The Government of the Russian Federation The Federal State Autonomous Institution of Higher Education National Research University Higher School of Economics Faculty of Business and Management School of Business Informatics Department of Higher Mathematics Algebra and Geometry Author: V.Goncharenko, associate professor Approved at the meeting of the Department Higher Mathematics 2018 Head of Department / A.A. Makarov/ Approved by the Academic Council of the Management and Digital Innovation educational program 2018 Chairman / S.G. Efremov/ Moscow, 2018

2 1. Course Description The program of the course describes the basic requirements for the knowledge and skills of students and determines the content and types of classes and assessment. The program is designed for lectures of this discipline, learning assistants and students enrolled in the program. The program is developed according to Educational Program of National Research University Higher School of Economics. The course aims to provide students with understanding of key concepts and methods of algebra and geometry for understanding the other practical courses, related to data analysis and programming. 2. Learning Objectives In the process of studying the discipline, students will become familiar with theoretical foundations and basic methods of solving tasks on the following topics Systems of linear equations. Row operations and Gaussian elimination. Vectors and Matrices. Linear spaces. Homogeneous systems and null space. Matrix inversion and determinants. Leontief input-output analysis. Complex numbers and their properties. Eigenvalues and eigenvectors. Diagonalization of matrices. Sequences, series and difference equations. Coupled first-order difference equations. Their applications in economics and finance Inner product and orthogonality. Lines in, lines and hyperplanes in n R. 2 R, planes and lines in Orthogonal diagonalisation. Quadratic forms and conic sections 3 R

3 Direct sum and projections. Fitting function to data: least squares approximation 3. Learning Outcomes As a result of this course a student will: know main definitions and results of algebra and geometry to be essential for understanding further math courses and practical courses, related to data analysis, programming, economic theory and corporate finance; be able to formalize the problem from subject area, choose the adequate methods of solutions, perform essential calculations and to interpret the results; have essential skills of solving problems to be important in professional activity. 4. Course Plan Theme Total Hours Self Lecture hours Seminar hours study Systems of linear equations. Row operations and Gaussian elimination. Vectors and Matrices. Vector spaces. Homogeneous systems and null space. Determinants and matrix inversion. Leontief input-output analysis. Complex numbers and their properties Eigenvalues and eigenvectors. Diagonalisation of matrices. First module (40 hours) Second module (40 hours)

4 National Research University «Higher School of Economics» Sequences, series and difference equations. Coupled first-order difference equations. Their applications in economics and finance. Inner product and orthogonality. 2 3 Lines in R, planes and lines in R n, lines and hyperplanes in R. Orthogonal diagonalisation. Quadratic forms and conic sections. Direct sum and projections. Fitting function to data: least squares approximation In total Detailed course content Theme 1. System of linear equations and matrices. Introduction to Systems of Linear Equations. Gaussian Elimination. Matrices and Matrix Operations. Algebraic Properties of Matrices. Powers of the ma- 1 trix. Transpose matrix. Inverse matrix. Method for finding A - with row operations. Symmetric matrices. Vectors in n R. Inner product. (Ch 3, Ch 4: p. 4.1). Applications (Ch 1). Theme 2. Vector spaces and Homogeneous systems. Real Vector Spaces and Subspaces. Linear Independence and Dependence of vectors. Coordinates and Basis. Dimension. Solution Spaces of Homogeneous Systems. Change of Basis. Row Space, Column Space, and Null Space. Rank, Nullity and the Fundamental Matrix Spaces.

5 (Ch 13-16). National Research University «Higher School of Economics» Applications (Ch 4). Theme 3. Determinants and inverse matrix. Determinants of matrices. Finding determinants by Cofactor Expansion. Evaluating Determinants by Row Reduction. Properties of Determinants. Cramer s Rule. Nondegenerate matrix and existence of inverse. Adjoint matrix. Using adjoint matrix to find inverse matrix. Leontief input-output analysis. (Ch 3). Applications (Ch 2). Theme 4. Complex numbers. Complex numbers. Complex conjugate. Algebra of complex numbers. The complex plane. The polar form of a complex number. The modulus and the argument of a complex numbers. Complex vector spaces and complex matrices. (Ch 13). Applications (Ch 5: p. 5.3). Theme 5. Eigenvalues and Eigenvectors.

6 Eigenvalues and Eigenvectors. Diagonalization of a square matrix. Eigenvalues and Eigenvectors of Matrix Powers. Determinants and eigenvalues. Similar matrices. Finding the power of a matrix using diagonalization. (Ch 8). Applications (Ch 5). Theme 6. Difference equations. Sequences and progressions. Compound interest. Frequent compounding Series and financial applications. First-order difference equations and their solution. Long-term behavior. The cobweb model. Second-order difference equations. Behavior of solutions. Economic applications. (Ch 9). 2. R Anthony, M. and N. Biggs. Mathematics for Economics and Fnance: Methods and Modelling (Ch 3, 4) Theme 7. Euclidean vector spaces. Lines, planes and hyperplanes. Inner product and orthogonality. Euclidean vector spaces. Lines in planes and lines in 3 R, lines and hyperplanes in 2 R, n R. Geometry of linear systems. (Ch 1, sections ). Applications (Ch 3).

7 Theme 8. Quadratic forms and conic sections. Orthogonal diagonalization of symmetric matrices. Quadratic forms. Quadratic forms and conic sections. Circle, ellipse or hyperbola. (Ch 11). Applications (Ch 7, section ). Theme 9. Direct sum and projections. Fitting function to data. The direct sum of two subspaces. The orthogonal complement of a subspace. Orthogonal complements of null spaces and ranges. Projections. Orthogonal projections. Orthogonal projection onto the range of a matrix. Minimizing the distance to a subspace. Fitting functions to data: least squares approximation. (Ch 12). Applications (Ch 6, section ). 5. Reading list a) Required. Cambridge University Press, Applications. John Wiley & Sons Asia Plc Ltd, 2014, 11th edition. 3. R Anthony, M. and N. Biggs. Mathematics for Economics and Finance: Methods and Modelling. Cambridge: Cambridge University Press, 1996.

8 Inc., National Research University «Higher School of Economics» b) Optional 1. R Lay, D.C. Linear Algebra and its Applications. Pearson Education, 2. Красс М. С., Чупрынов Б.П. Основы математики и ее приложения в экономическом образовании: Учебник. М.: Дело, Беклемишев Д.В. Курс аналитической геометрии и линейной алгебры: Учебник. М.: Высшая школа, Бугров Я.С. Никольский С.М. Элементы линейной алгебры и аналитической геометрии: Учебник для вузов. М.: Наука, Бурмистрова Е.Б., Лобанов С.Г. Линейная алгебра с элементами аналитической геометрии: Учебное пособие. М.: Изд-во ГУ-ВШЭ, Солодовников А.С., Бабайцев В.А., Браилов А.В. Математика в экономике: Учебник. В 3-х ч. Ч.1. М.: Финансы и статистика, Simon C.P., Blume Z. Mathematics for Economists. W.W. Norton and Company, Grading System The final grade can be obtained by rounding the score S obtained by the following formula: S=0,5*A+0,5*E, where E is a mark for the final exam on the course, held at the end of the second module (duration is 120 minutes) and A is an accumulated mark. The accumulated score A for both modules is obtained by rounding the score obtained by the following formula: 0.3 * C1+0.3* C2+0.4*W, where C1 and C2 are grades for the first (held at the end of the first module or at the beginning of the second one) and the second (held at the middle of December) control works. W is score obtained for the regular quizzes held at seminars, homeworks and seminar activity.

9 The rewriting of the control works at extra time is not allowed. 7. Guidelines for Knowledge Assessment Examples of control tasks м x1 + 3x2 + 6x3 = 40, 1. Solve the system of linear equations п н3x x x3 = 130, п по 2x1 + 3x2 + 10x3 = Find the general solution of the system of linear equations м x1 + x2 + x3 + x4 = 4, п н - x1-2 x3 = - 5, п по - 2x1-3x2-2x4 = ж 1-1ц 3. Find the matrix з зи - 1-1шч. ж ц Find the determinant of the matrix зи ш ч ж ц з з From the rows system of matrix A = з select the з- - зи ш ч maximal linearly independent subsystem and express the remaining rows as a linear combination of the selected ones. ж 2-5 3ц 1 6. Find the inverse matrix A - if A = and check the condi- зи 3-2 3шч - 1 tion AЧ A = E is valid. 7. Consider an economy with three industries, i 1 : water, i 2 : electricity and i 3 : gas interlinked so that the corresponding consumption matrix is ж ц C = Each week the external demands for water, electricity and gas з зи шч are, respectively, d 1 = 40000, d 2 = , d 3 = (units measured in dollars). (a) How much water, electricity and gas is needed to produce a unit of electricity? (b) What should be the weekly production of each industry in order to satisfy all demands exactly?

10 Solve the matrix equation ж - - ц X ж - = ц з и 3 5 чш зи 5-3 2шч. 9. Consider the complex numbers z = 3 - i, w = 1+ i and 6 ( 3 - i) q =. Plot z and w as points in the complex plane. Express them in exponential form and hence evaluate q. Express q in the form a + ib. 10 ( 1+ i) Find eigenvalues and eigenvectors of the matrix C = ж з - ц зи шч. 11. Find the Cartesian equation of the plane passing through the point x+ 4 y z+ 1 A( 1, - 1, 4) which is orthogonal to the line = = Find the Cartesian equation of the median AM of the triangle ABC with vertices A( - 5,3,1). B (10,9,4), C( - 20, - 15, 6). 13. Find the Cartesian equation of the plane, all points of which are equidistant from the points A( - 5,2, - 2) и B( - 15, - 10, - 14). 2 2 r T r 14. Express the quadratic form 9x + 4xy + 6 y as u Au, where A is a r symmetric 2 2 matrix and u = ( x, y), and find the eigenvalues of A. Deduce whether the quadratic form is positive definite or otherwise, and determine what 2 2 type of conic section is given by the equation 9x + 4xy + 6 y = 10. Orthogonally diagonalize the matrix A and use this information to sketch the curve 2 2 9x + 4xy + 6 y = 10 in the xy-plane Suppose that vector space V = R with the standard inner product r r r r S = Lin u, v u = 1;2; 1 v = 1;0;1. and assume that subspace { } Describe S. where ( ) 16. Quantities X and Y are related by a rule of the form and ( ) a Y = + b for X some constants a and b. Use the following data to estimate a and b by the least squares method: X Y 1/ 5 1/ 4 1/ 3 1/

11 8. Methods of Instruction Delivery of homework can be done remotely by s. Also examples of current, intermediate and control work s tasks can be given on the teacher's page and in LMS system. The result of essential homework, quizzes and final control work are sent by . Every week students get the brief summary of the lectures to help them to do homeworks and literature study.