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Бакалаврская программа «Социология и социальная информатика»

Algebra and Analysis

2021/2022
Учебный год
ENG
Обучение ведется на английском языке
6
Кредиты
Статус:
Курс обязательный
Когда читается:
1-й курс, 1, 2 модуль

Преподаватель

Course Syllabus

Abstract

The course of Algebra and Analysis is intended for beginners. Its goal is to introduce the students to the language of mathematics and basic ideas of graphs, vectors, matrices, derivation, and integration. These are indispensable tools of any domain of science using data analysis. 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

Learning Objectives

  • The goals of the course are to introduce the students to basic ideas of graphs, vectors, matrices, derivation, and integration
  • to accustom students to the language of mathematics, to develop theirs abilities to communicate with mathematical symbols and logic quantifiers
  • to develop the ability of students to prove theorems or/and formulas of the course by themself and to demonstrate them the importance of rigorous proofs
  • to overcome fear to read math books and articles
  • to make a habit of mathematical rigor in scientific discussions
Expected Learning Outcomes

Expected Learning Outcomes

  • applies basic facts from graph theory in simple social networks
  • uses the mathematical induction for proofs
  • performs computations with matrices, vectors, linear systems
  • applies the matrix transformations and decompositions
  • analyzes functions by construction of theirs plots
  • performs actions with vectors in a linear space, explains geometric meaning of the operations
  • uses equations of the line and the plane to compute distances
Course Contents

Course Contents

  • "Set theory. Logic. Combinatorics. Graphs."
    Logic quantifiers. Euler-Venn diagrams. Product of sets. Inclusion-Exclusion formula. De Morgan's laws. Maps: injection, surjection, bijection. Composition of functions. Method of math induction. Newton's binomial formula. Stable matchings: Gale-Shapley algorithm. Graphs. Adjacency matrix. Theorem about the sum of vertices degrees. Theorem about the number of paths of length N from a vertex to a vertex. Bipartite graph. Random walk on a graph. Page-Rank formula.
  • “Linear Algebra”
    Metric space. Examples of metrics and similarity measures. Example: embeddings of words, recommender systems. Vectors. Addition of vectors. Multiplication of a vector by a number. Sum of vectors. Angle between vectors. Scalar product. Cauchy-Bunyakovsky-Schwartz inequality. Linear combination and linear span of vectors. Determinant is a volume. Properties of determinant. Rank of a matrix, geometric interpretation. Systems of linear equations. Gauss elimination method. Reduced echelon form of a matrix. Inverse matrix. Matrix representation of a linear system. Existence and uniqueness of solution of a linear system. Linear mapping. Basis of a Linear space. Coordinates of a vector in a basis. Computation of coordinates in different basis, its connection with the linear systems. Eigenvalues. Eigenvectors. Geometric interpretation of eigenvectors. Eigendecomposition. Idea of SVD decomposition and dimesnsionality reduction.
  • "Analytic geometry"
    The general equation of a line on a plane. The condition of parallelism and perpendicularity of lines. The distance from a point to a line. The general equation of the plane. The condition of parallelism and perpendicularity of planes. The equation of a line in space.
  • ''Mathematical Analysis''
    Physical and geometric meaning of the derivative. Monotonicity. Convexity. Local and global extrema. Types of discontinuities. Asymptotes equation. Tangent line equation. Taylor decomposition of a function. Definite integral. Integration as the area. Darboux sums. Primitive of a function. Fundamental theorem of calculus. Integration by parts. Indefinite integrals. Linear regression formula derivation. Convergent, divergent sequences. Infinitely small, infinitely large values. Typical indeterminate forms.
Assessment Elements

Assessment Elements

  • non-blocking Activity grade
    At the seminars that pass online, all students must keep their cameras switched on.
  • non-blocking Exam
    At the exam it is forbidden to use any notes and any copybooks, as well as any gadget with internet. The students who violate these rules will get a warning, if they violate the rules twice they will get 0. All students must switch on cameras on their computers during all the exam, if the exam passes online. They must stay clearly visible by the cameras, if the exam pass online. Otherwise, the mark will be 0.
  • non-blocking Average of all Mini-tests
    This mark is constructed as an average of Mini-tests. The duration of each mini-test is 30 minutes. Mini-tests are given more frequently than usual tests. They are used for more regular control of students knowlege and unerstanding. The rounding to the nearest integer is used. At the tests that pass online, the students must switch on theirs cameras.
  • non-blocking Test 1
    All students must write the Tests and Mini-tests at the time that the teacher announced in advance. Those who are absent at the Tests by legitime reasons (medical reasons confirmed by the corresponding medical documents), they will have an opportunity to write the test at the end the course. If a student was absent at the tests without legitime reasons he/she gets 0 for this test. At the tests that pass online, all students must keep cameras switched on. Otherwise, the mark will be 0.
  • non-blocking Test 2
    All students must write the Tests and Mini-tests at the time that the teacher announced in advance. Those who are absent at the Tests by legitime reasons (medical reasons confirmed by the corresponding medical documents), they will have an opportunity to write the test at the end the course. If a student was absent at the tests without legitime reasons he/she gets 0 for this test. At the tests that pass online, all students must keep cameras switched on. Otherwise, the mark will be 0.
  • non-blocking Self-study report
  • non-blocking Individual Home Test
    For this test students can use any resourses they could find, including internet sites, internet calculators, and programming
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.1 * Activity grade + 0.15 * Average of all Mini-tests + 0.25 * Exam + 0.1 * Individual Home Test + 0.1 * Self-study report + 0.15 * Test 1 + 0.15 * Test 2
Bibliography

Bibliography

Recommended Core Bibliography

  • F. Aleskerov, H. Ersel, D. Piontkovski. Linear Algebra for Economists. Springer, 2011
  • 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

Recommended Additional Bibliography

  • Алескеров Ф.Т., Хабина Э.Л., Шварц Д.А. - Бинарные отношения, графы и коллективные решения - Издательство "Физматлит" - 2012 - 344с. - ISBN: 978-5-9221-1363-2 - Текст электронный // ЭБС ЛАНЬ - URL: https://e.lanbook.com/book/59762