# 2 semester

1

Name of the course

Algebra

2

Year of study, speciality

1,

Mathematics (Science-Design Activity)

3

Semester of study

2

4

Credits

4

5

Lecturer

Bondarenko A.A.

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Course objectives

To acquaint students with fundamental methods of general and linear algebra, with basic algebraic structures – groups, rings and fields; to create a base for studying basic concepts and methods of modern mathematics; to form mathematical thinking among students; to acquaint with methods of mathematical proofs; to study algorithms for solving specific mathematical problems. To instill in 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, to extract roots of complex numbers, use de Moivre formula;

– calculate determinants;

– perform operations on matrices;

– solve systems of linear equations;

– find the basis of a vector space, of a sum and intersection of subspaces, coordinates of a vector in a given basis, find the rank of the matrix and vector systems;

– find eigenvalues and eigenvectors of a matrix and a linear operator;

– reduce the quadratic form to the canonical form;

– reduce the orthogonal operator to the canonical form;

– – find an orthonormal basis, an orthogonal complement to the subspace.

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Prerequisites

Algebra (part 1)

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Contents

Vector spaces

The definition and examples. A system of generators, finite-dimensional spaces. A linear dependence of vectors.

A basis, a dimension. The coordinates of a vector, their change when changing the basis. A transition matrix.

The rank of a system of vectors. The rank of a matrix. Subspace, its dimension.

The sum and intersection of subspaces, their dimensions. A direct sum of subspaces.

Linear equation systems

Systems of linear equations, homogeneous systems. Kronecker-Capelli Theorem. The fundamental system of solutions.

The structure of the set of solutions of an arbitrary system of linear equation.

Linear operators of vector spaces

The linear operator, its kernel and image. The rank and defect. The matrix of a linear operator. The change in the matrix of the operator on going to another basis.

Algebraic operations on linear operators. The matrix of a composition and a sum of linear operators. The conditions of invertibility of the operator. Invariant subspaces. The restriction of an operator to an invariant subspace.

The matrix of the operator under the existence of an invariant subspaces and when the space is decomposed into a direct sum of invariant subspaces. Eigenvalues and eigenvectors of the operator.

Normal forms of matrices

The characteristic polynomial of the matrix. Definition and construction of Jordan’s normal form (without proof).

Minimal polynomial. The diagonalizability criterion of a matrix over a field. Normal form of Frobenius (without proof).

Linear, bilinear and quadratic forms on vector spaces.

A matrix of a bilinear and a quadratic form, its change by changing the basis.

Orthogonality with respect to bilinear form. Decomposition into a direct sum of orthogonal subspaces.

Canonical form. Reduction of the quadratic form to the canonical form. A normal form of real and complex quadratic forms.

Euclidean and unitary spaces.

 Sesquilinear functions. A definition of an Euclidean and unitary spaces. The length of the vector, the angle between the vectors. Orthogonal and orthonormal bases. The process of Gram-Schmidt orthogonalization.

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### RecommendedLiterature

 1. Milovanov M.V, Tyshkevich R.I, Fedenko A.S. Algebra and analytic geometry. V. 1. Minsk: Amalfeya, 2001. 2. Milovanov M.V, Tolkachev M.M, Tyshkevich R.I, Fedenko A.S. Algebra and analytic geometry. V. 2. Minsk: Amalfeya, 2001. 3. Burdun A.A, Murashko E.A, Tolkachev M.M, Fedenko A.S. A collection of problems on algebra and analytic geometry. Minsk: Universitetskoe, 1999. 4. Monakhov V.S, Buzlanov A.V. Algebra and Number Theory: Workshop. Minsk: Izd. center of BSU, 2007. 5. Proskuryakov I.V. Problems in linear algebra. Moscow: Nauka, 1978. 6. Faddeev D.K, Sominsky I.S. Collection of problems on higher algebra. Moscow: Nauka, 1977. 7. Barkovich O.A. Algebra: problems for practical studies and independent work.  Part 1. Introduction to algebra. Minsk: BSPU, 2005. 8. Barkovich O.A. Algebra: problems 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. Advanced reading: 16. Ireland K., Rosen M. Classical introduction to the modern theory of numbers. Moscow: Mir, 1987. 17. Van der Waerden B. L. Algebra. Moscow: Nauka, 1976. 18. Kostrikin A.I, Manin Yu.I. Linear Algebra and Geometry. Moscow: Nauka, 1983. 19. Kargapolov M.I, Merzlyakov Yu.I. Fundamentals of group theory. Moscow: Nauka, 1972. 20. Lang S. Algebra. Moscow: Mir, 1968. 21. Collection of problems in algebra. Ed. A.I. Kostrikin. Moscow: Nauka, 1987.

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Teaching methods

Verbal, visual, problem-based, practical, dialog-based and heuristic.

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Language of teaching

Russian

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Conditions (requirements), current control

– tests.

examinations marks are given taking into account:

40% – semester work,

60% – oral answer in an examination

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Form of current assessment

an examination, control test