1 
Name of the course 
Algebra 

2 
Year of study, speciality 
1, Mathematics (ScienceDesign Activity) 

3 
Semester of study 
1 

4 
Credits 
4 

5 
Lecturer 
Bondarenko A.A. 

6 
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. 

7 
Prerequisites 


8 
Contents 
Arithmetic of integers, complex numbers The theorem on division with remainder for integers. Euclidean algorithm. Definition of complex numbers, conjugate complex numbers. Trigonometric form of a complex number, Muavr formula, geometry of operations on complex numbers. Nth degree roots of a complex number, nth degree roots of unity, primitive roots and their properties. Matrices and operations on matrices Rectangular matrices, equality of matrices, matrix addition and multiplication of a matrix by scalar, transposition of a matrix. Matrix multiplication, associativity of matrix multiplication, the relationship between the operations of addition, multiplication and transpose of matrices. Permutations, substitutions. Determinants and their application The number of permutations of a finite set, the parity of the permutation, the number of even (odd) permutations of a finite set. The number of substitutions of a finite set, the parity of the substitutions, the decomposition of the substitutions into the product of independent cycles. Definition of the determinant and its properties. Laplace’s Theorem. Construction of the inverse matrix, Cramer rule. Polynomials in one variable The definition of a polynomial of one variable, equality of polynomials, the theorem of division with remainder, Roots of polynomials, multiple roots, rational roots of polynomials with integer coefficients. The main theorem of the algebra of complex numbers, the Vieta formulas, irreducible polynomials over Q, R, and C. Algebraic operation, concept of a group, a ring, a field. Definition of binary algebraic operation, neutral and symmetric elements. Definitions and examples of groups, rings, fields and their properties.


9 
RecommendedLiterature



10 
Teaching methods 
Verbal, visual, problembased, practical, dialogbased and heuristic. 

11 
Language of teaching 
Russian 

12 
Conditions (requirements), current control 
– check of individual tasks, – tests. examinations marks are given taking into account: 40% – semester work, 60% – oral answer in an examination 

13 
Form of current assessment 
an examination, control test 