Infopaket

Algebra and Geometry

Обов’язкова дисципліна для ОП

First

2021/2022

2 Semester

12

The student will have the ability of abstract arguing on problems of analytical geometry and linear algebra. They will obtain knowledge on applications of systems of linear equations in various nu-merical problems and on applications of methods of analytical geometry and linear transformations in analytic and graphic problems. The student will know how to study and master modern tech-niques. A successful pupil will formulate and investigate correctly mathematical results, particularly in discrete analysis, find standard approach for solving theoretical and numerical problems. They will have the knowledge on unique and multiparameter family of solutions on examples of systems of linear equations.

Upper secondary education.

At the beginning of the program we consider systems of vectors and systems of coordinates on plane and in space. Then we introduce basic properties of conic sections, lines, planes in a coordinate space. In the next section we give the notion of a matrix of systems of linear equations, its determinants and their properties. Then we define abstract linear space and linear transformations on them. To proper study of linear operators it considers basic properties of polynomials in one variable and complex numbers. Then we describe eigenvalues and eigenvectors of an endomorphism and Jordan canonical form of the matrix of an operator. In the last part of the course we introduce Euclidean space, Hermitian operator, polar and SVD decompositions. And at the end we consider quadratic functions and their properties. To be able to remember theoretical knowledge, students solve various numerical problems.

1. Yefimov N.V Kratkiy kurs analiticheskoy geometrii. M.: “Nauka”, 1969. – 272s. 2. Charín V.S. Líníyna algebra. K: “Tekhníka”, 2003. 3. Kostrikin A.I. Vvedeniye v algebru, M: Fizmatlit, 2000. 4. Strang G. Linear algebra and its applications. Andover: “Cengage learning”, 2006. 5. Kletenik I.V. Sbornik zadach po analiticheskoy geometrii. M.: “Nauka”, 1987. – 724s. 6. Proskuryakov I.V. Sbornik zadach po lineynoy algebre . Sankt-Peterburg: “Lan'”, 2021. 7. Faddeyev D.K., Sominskiy I.S. Zadachi po vysshey algebre. . Sankt-Peterburg: “Lan'”, 2008. 8. Bezushchak O. O., Ganyushkív O. G., Kochubíns'ka Ê A.. Navchal'niy posíbnik íz líníynoí̈ al-gebri. K.: VPTS «Kií̈vs'kiy uníversitet», 2019. 9. Il'in V.A., Poznyak E.G. Lineynaya algebra. M: FML, 2005. 10. Marinich O. V., Proskurín D. P. Skínchennovimírniy líníyniy analíz. Teoríya viznachni-kív (∆), K: «Tsentr navchal'noí̈ líteraturi», 2014. 11. Travkín YU. Í. Líníyna algebra í analítichna geometríya. KH.: «Maydan», 2009.

40 lectures, 3 consultations , 42 practical lessons. During a lecture: additional discussion on application problems of coordinate method and linear algebra methods, direct answers to questions. At the time of practics: solving of tipical problems, discussions on applications, use of morden computers and software in order to find numerical characteristic of geometric systems, matrices, polynomials and operators.

Full set of problems and theoretical questions are prepared for 6 modules (class tests) and 2 examinations. The set cover all themes of the course and uses almost all methods from lectures. The score of successful student obtained by all the test’s and exam’s assessment is at least 60 % of course grade.

Ukrainian