Бакалавриат 2017/2018

Алгебра и анализ

Статус: Курс по выбору (Социология и социальная информатика)

Направление: 39.03.01. Социология

Кто читает: Департамент математики

Где читается: Санкт-Петербургская школа социальных наук

Когда читается: 1-й курс, 1, 2 модуль

Формат изучения: без онлайн-курса

Преподаватели: Калинин Никита Сергеевич, Маринов Илья Львович

Язык: английский

Кредиты: 6

Контактные часы: 70

Course Syllabus

Abstract

The course of Algebra and Analysis (modules 1, 2) is intended for beginners. Its goal is to introduce the students to the language of mathematics and basic ideas of vectors, matrices, derivation, and integration. These are indispensable tools of any domain of science using data analyses. Special attention will be devoted to applications. This course will help you to gain a higher level of mathematical maturity necessary in subsequent courses.

Learning Objectives

• Its goal is to introduce the students to the language of mathematics and basic ideas of vectors, matrices, derivation, and integration.

Expected Learning Outcomes

• can understand the language of the set theory, foundations of analyses
• knows the concepts of graph theory, can use the method of mathematical induction
• can calculate the derivative of a function of one variable and examine the function of one variable
• can perform actions with vectors in space, calculate the distance between lines and from point to line
• able to perform actions with matrices and determinant
• can solve systems of linear equations
• can perform actions with vectors, eigenvectors and linear operators

Course Contents

• Basic set theory and functions
  Sets, intesections, unions, power, Euler-Venn diagrams, Newton’s binomial Graphs, random walks on them via adjacency matrix, Inclusion-exclusion formula, Gale-Shapley algorithm
• Combinatorics
  Elementary concepts of graph theory. Adjacency matrix, the number of paths in the graph. Random walks on a graph. The method of mathematical induction, the inclusion-exclusion formula, Newton’s bin. Stable matching. Gale-Shapley Algorithm.
• Differential calculus and its applications
  The concept of a derivative function of one variable. Geometric and economic interpretation of the derivative. The tangent equation. The derivative of a sum, of a product, of a particular, complex, and inverse function. Derivatives of basic elementary functions. The concept of extrema of a function of one variable. The task of maximizing the profits of the company. Local extremum (internal and boundary) functions of one variable. A necessary condition for an internal local extremum (Fermat's theorem). Mean value theorems (Rolle, Lagrange, and Cauchy theorems) and their geometric interpretation. The rule of Lital. Taylor and Maclaurin formulas and their use for the representation and approximate calculation of function values. A sufficient condition for the strict increase (decrease) of the function on the interval. Sufficient conditions for a local extremum of a function of one variable. Convex (concave) functions of one variable. A necessary and sufficient condition for convexity (concavity). Vertical and non-vertical asymptotes of a graph of a function of one variable. Investigation of the function of one variable using the first and second derivatives and plotting it. Determination of the global maximum (minimum) of the function of one variable in the field of its definition. Examples of sequences. The limit of the numerical sequence. The existence of a limit in a limited monotone sequence. The nested segment lemma. Subsequences. Bolzano-Weierstrass theorem on the allocation of a convergent subsequence. The linear regression formula.
• Analytic geometry
  Rectangular coordinate system on the plane. Distance between points. The division of the segment in this regard. Vectors. Equality vectors. The coordinates of the vector. Addition of vectors. Multiplication of a vector by a number. Decomposition of a plane vector in two noncollinear vectors. Scalar product of vectors. The general equation of a line on a plane. The condition of parallelism and perpendicularity of straight lines. Parametric and canonical equations of the line. The distance from a point to a line. Transformation of coordinates of a point when replacing a coordinate system. Decomposition of a vector into three non-coplanar vectors. Vector product of vectors. Mixed product of vectors. The general equation of the plane. The condition of parallelism and perpendicularity of planes. The equation of a line in space. Mutual arrangement of a straight line and a plane, two straight lines.
• Matrices and determinants
  Matrices. Matrix and extended matrix of a system of linear equations. elementary transformations of matrices. Reversibility of elementary transformations. Reduction of matrices to a stepwise form by elementary transformations. Gauss method for solving systems of linear equations. The solution of systems of linear equations with a step matrix of the system. The general solution of systems of linear equations. The main and free ¬ unknown. Geometric interpretation of systems of linear equations in the case of two or three unknowns. Nonzero solutions of a homogeneous system of equations. Determinant and elementary transformations. Construction of a determinant by column expansion. The determinant of the transposed matrix. Calculation of the determinant by line expansion. Sum of matrices. Multiplication of a matrix by a number. The product of matrices. Matrix recording of a system of equations. Properties of arithmetic operations on matrices. Inverse matrix and Cramer formulas. Construction of the inverse matrix by elementary transformations. Transformation of coordinates when changing the basis. The simplest consequences of the axioms of linear space. The subspace of linear space. The simplest properties of linearly dependent vectors. The basis and coordinates of vectors. The existence of a basis of finite-dimensional space. The dimension of linear space.
• Systems of linear equations
  The rank of the matrix. Grade stepped matrix. Invariance of a rank at elementary pre-transformations. Matrix rank theorem. Criterion for linear independence of a row (column) system. The rank of the product of the matrices. The determinant of the product of matrices. Vector notation of a system of equations. Kronecker-Capelli theorem on the compatibility of a system of linear equations. The dimension of the space of solutions of a homogeneous system of linear equations. The structure of the set of solutions of a system of linear equations. Theorem on the choice of principal and free unknowns.
• Vector algebra
  Scalar product. Cauchy-Bunyakovsky inequality. Inequality of the triangle. The length of the vector and the angle between the vectors. Orthogonality of vectors. Eigenvectors of the matrix. Transformation of coordinates of a point when replacing a coordinate system. Linear mappings. Linear operators associated with linear mappings.

Assessment Elements

• Test 1
• Test 2
• Test 3
• Class participation
• Exam
• Individual homework

Interim Assessment

• Interim assessment (2 module)
  0.06 * Class participation + 0.4 * Exam + 0.06 * Individual homework + 0.162 * Test 1 + 0.162 * Test 2 + 0.156 * Test 3

Bibliography

Recommended Core Bibliography

• Gareth Williams. (2012). Linear Algebra with Applications. [N.p.]: Jones & Bartlett Learning. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=459020

Recommended Additional Bibliography

• Fuad Aleskerov, Hasan Ersel, & Dmitri Piontkovski. (2011). Linear Algebra for Economists. Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.b.spr.sptbec.978.3.642.20570.5