2 semester

1

Name of the course

Algebra and Number Theory (Part 2)

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Year of study, speciality

1, Mobile Devices Software

3

Semester of study

2

4

Credits

4

5

Lecturer

Shlyk V.A.

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Course objectives

Teaching students fundamental methods of general algebra, linear algebra, number theory; acquaintance with basic algebraic structures – groups, rings and fields; creation of a base for studying basic concepts and methods of modern mathematics; formation of mathematical thinking among students; familiarity with methods of mathematical evidence; studying algorithms for solving specific mathematical problems. Teaching students the ability to independently study educational and scientific literature in the field of mathematics.

As a result of studying the course, a student should be able to:

  • perform operations with complex numbers in algebraic and trigonometric form, extract roots of complex numbers, apply De Moivre’s formula;
  • calculate determinants;
  • perform operations with matrices;
  • solve systems of linear equations;
  • find a basis of a vector space, the sum and intersection of subspaces, the coordinates of a vector in a given basis, find the rank of a matrix and system of vectors;
  • find the eigenvalues ​​and eigenvectors of a matrix and linear operator;
  • reduce a quadratic form to the canonical form;
  • reduce the matrix of an orthogonal operator to the canonical form;
  • find an orthonormal basis, an orthogonal complement of a subspace;
  • determine whether a given subset is a subgroup of a group, a subring or an ideal of a ring, a subfield of a field;
  • perform calculations in the factor group, factor ring

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Prerequisites

Algebra and Number Theory (Part 1)

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Contents

Vector spaces

  • Definition and examples. System of generators, finite-dimensional spaces.
  • Linear independence of vectors. Steinitz exchange lemma. Basis, dimension.
  • Coordinates of a vector, their change when the basis is changed. Transition matrix, transformation of the coordinates of a vector. Subspace, its dimension.
  • The rank of a system of vectors. The rank of a matrix.
  • The sum and intersection of subspaces, their dimensions. The direct sum of subspaces.

Systems of linear equations.

  • Matrix form of a linear system. The Gauss method. Kronecker-Capelli theorem. Homogeneous systems, the condition for existence of a nontrivial solution.
  • A fundamental system of solutions. Connection between solutions of homogeneous and non-homogeneous systems.
  • The subspace defined by a system of linear equations.
  • Differentiable functions in several variables
  • Linear map, its core and image. Rank and defect. Algebraic operations on linear maps: sum, multiplication by a constant, composition.
  • Linear operator and its matrix, its change when the basis is changed.
  • Matrix of composition and sum of linear operators.
  • The space of linear operators and its connection with the space of matrices. Conditions for the invertibility of an operator.

Invariant subspaces. Eigenvectors and eigenvalues. Normal forms of matrices.

  • Restriction of an operator to an invariant subspace.
  • The matrix of an operator with an invariant subspace, under decomposition of the space into a direct sum of invariant subspaces.
  • Characteristic polynomial of an operator and matrix.
  • The Hamilton-Cayley theorem. An operator having a diagonal matrix in some basis; a condition of diagonalizability. Jordan matrix.
  • A theorem on the existence of a Jordan normal form of a matrix. Algorithm of reduction to the Jordan normal form. Frobenius  normal form.

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Recommended Literature

  1. Milovanov M.V, Tyshkevich R.I, Fedenko A.S. Algebra and analytic geometry. V. 1. Minsk. .: Amalfea, 2001.
  2. Milovanov M.V, Tolkachev M.M, Tyshkevich R.I, Fedenko A.S. Algebra and analytic geometry. V. 2. Minsk .: Amalfee, 2001.
  3. Burdun A.A, Murashko E.A, Tolkachev M.M, Fedenko A.S. A collection of problems on algebra and analytic geometry. Moscow: University, 1999.
  4. Monakhov V.S, Buzlanov A.V. Algebra and Number Theory: Workshop. Minsk: Izd.center of BSU, 2007.
  5. Proskuryakov I.V. Collected problems on linear algebra. Moscow: Nauka, 1974.
  6. Faddeev D.K, Sominsky I.S. Collection of problems on higher algebra. Moscow: Nauka, 1977.
  7. Barkovich O.A. Algebra: tasks for practical studies and independent work.  Part 1. Introduction to algebra. Minsk: BSPU, 2005.
  8. Barkovich O.A. Algebra: tasks for practical studies and independent work.  Part 2. Linear algebra. Minsk: BSPU, 2006.
  9. Kostrikin A.I. Introduction to Algebra. V. 1-3. Moscow: Phys.-Math. literature, 2000-2001.
  10. Gelfand I.M. Lectures on linear algebra. Moscow: MTsNMO, 1998.
  11. Kurosh A.G. The course of higher algebra. Moscow: Nauka, 1965 (and later editions).
  12. Maltsev I.M. Fundamentals of linear algebra. Moscow: Nauka, 1970.
  13. Faddeev D.K. Lectures on algebra. Moscow: Nauka, 1984.
  14. Vinberg E.B. Course of algebra. Moscow: Factorial-press, 2001.
  15. Vinogradov I.M. Fundamentals of number theory. Moscow: Nauka, 1976.

Additional literature

  1. Ireland K., Rosen M. Classical introduction to the modern theory of numbers. Moscow: Mir, 1987.
  2. Van der Waerden B. L. Algebra. Moscow: Nauka, 1976.
  3. Kostrikin A.I, Manin Yu.I. Linear Algebra and Geometry. Moscow: Nauka, 1983.
  4. Kargapolov M.I, Merzlyakov Yu.I. Fundamentals of group theory. Moscow: Nauka, 1972.
  5. Lang S. Algebra. Moscow: Mir, 1968.
  6. Collection of problems in algebra. Ed. A.I. Kostrikina. Moscow: Nauka, 1987.

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Teaching methods

Verbal, visual, problematic, practical, dialog-heuristic.

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Language of teaching

Russian

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Conditions (requirements), current control

  • verification of individual tasks,
  • colloquium,
  • test.

The score on the exam is set taking into account:

  • 40% – work in the semester,
  • 60% – oral answer in the exam.

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Form of current assessment

Examination, offset