CC BY
36416. Muminov M.I., Rasulov T.H. On the number of eigenvalues of the family of operator matrices // Nanosystems: Physics, Chemistry, Mathematics. 5:5 (2014). Рp. 619-626.
17. Rasulov T., Tretter C. Spectral inclusion for diagonally dominant unbounded operator matrices // Rocky Mountain J. Math., 2018. № 1. Рp. 279-324.
THE METHOD OF USING PROBLEMATIC EDUCATION IN TEACHING THEORY OF MATRIX TO STUDENTS Boboeva M.N.1, Rasulov T.H.2
1Boboeva Muyassar Norboevna - Assistant, DEPARTMENT OF THEORY OF PRIMARY EDUCATION;
2Rasulov Tulkin Husenovich - PhD in Mathematics, Head of Department, DEPARTMENT OF MATHEMATICS, BUKHARA STATE UNIVERSITY, BUKHARA, REPUBLIC OF UZBEKISTAN
Abstract: the following article firstly deals with brief overview of theory of matrix. The issue of the relevance of problem-based education in the teaching of mathematics in higher education institutions was discussed as well. Examples of problem solving using elements of matrix theory are given. In the first problem, the problem of solving a matrix equation was brought to the solution of a system of linear equations. In the second problem, the problem of determining the order of the determinant corresponding to the matrix and the sign of the expression using the given expression has been analyzed.
Keywords: problematic education, matrix theory, methods of teaching, higher education, matrix equation, determinant.
In the current rapid development world, Mathematics plays an important role in training intelligent, creative thinking and independent decision-maker students. It is well known to us that the mathematical training of students must provide a theoretical basis for the study of other natural-scientific, general and specialized disciplines. For fulfilling this, it is important to teach students the elements of matrix theory using a variety of interactive methods.
It is well known that the first problem of linear algebra is the problem of linear equations. In the process of solving such equations, the concept of determinant emerges. As a result of studying the system of linear equations and their determinants, the concept of matrix was introduced. The introduction of the concept of color of the matrix by G. Frobenius allows finding the conditions for the solution of a system of linear equations. The concept of the matrix was introduced by James Joseph Sylvester in 1850 year. Initially, the matrix was developed in connection with the replacement of geometric objects and the solution of linear equations. Nowadays, matrices are one of the most important applied tools of Mathematics.
Matrices are widely used in various fields of Mathematics, Engineering and Economics. For example, they are used in Mathematics to solve systems of algebraic and differential equations, in quantum theory to predict physical quantities, and in the construction of modern aircraft in aviation. That is why, it is important to provide students with detailed information about matrix theory.
A variety of interactive methods [1-6] can be used to teach matrix theory to students. In this article, we have suggested to focus on the advantages of problem-based learning in teaching this theory.
Problem-based learning is a learning process based on solving problem situations. Problems and examples in the theoretical subject materials studied in Mathematics in Higher education institutions can be divided into problematic and non-problematic types according to their content. If the problem-solving process in the study material contains new mathematical concepts, facts, and rules for students that cannot be solved by the previous method, but requires new methods of
solution, then such a problem or example is problematic in content, and vice versa, the problem or examples can be given by the teacher to the students to solve, such problems and examples would not be problematic for the students; because they gain material without working independently and searching for new ways to solve the problem; they learn only from teachers and examples will be differed only in coefficients from previous.
Here are examples of how to generate a problem situation using elements of matrix theory.
As a result of studying problematic situations, one can always be sure that the problem cannot be clearly studied in advance. AX=C, XB=C, AXB=C matrix equations are usually solved using the inverse matrices of the both sided A,B matrices on either the left, right sides of the unknown X matrix. But the given matrices do not necessarily have to be inversed. For example,
f-10 6^
f!0 - 6 ^
B
V 5 - 3 J
If
f10 - 6 >
X =
V5 - 3 J
c =
v- 30 18 J
v- 30 18,
We cannot use inverse matrices in finding the solution of a matrix equation because there is no inverse matrix to it, because the rows of the matrix are linearly connected. But this matrix equation has a solution.
f
For finding the solution of X =
-X,
V X3
-
\
X
matrix B , it is necessary to solve the
4 J
following system of equations, which is formed by multiplying B matrix and equating the result C to the matrix:
10 Xj + 5x2 =-10
- 6xj - 3X2 = 6 10 X3 + 5 X4 =-30
- 6X3 - 3X4 = 18
f
We find the solution of the following equation system X =
X
V X3
- 2 - 2 Xj ^
- 6 - 2X3 j
, X3 G R .
It can be seen from the given example that the answer found to one question in the process of solving the equation gave rise to another question, and so on.
The answer to questions such as when a matrix equation has a single solution and in what cases there are an infinite number of solutions stems appear from the answers found to the questions. In the papers [7-19] it was derived the well-known Faddeev's matrix equation for the eigenfunctions of operator matrices.
Now we consider the question of determining a13a21a32a41a54a65a16 and aisas2a46a63a35a71as4a27, and clarify what order determinant of expressions is involved
in the calculation and with what sign.
A student who knows the definition of a determinant is required to carefully study the pairs of natural numbers involved in the indices of expressions. The student, who remembers that the first number in pairs represents the row order of the matrix, which is the determinant, and the second represents the column order, determines how many rows and columns are involved to form the
<
X
given expressions. He also knows that the determinant is a number that corresponds to a square matrix, and checks that each row and each column in the expressions present only once.
The first of the given expressions consists of seven multipliers. There are no repetitions in the first and second indices. Hence, this expression is involved in the calculation of the seventh-order square matrix. To determine which sign is involved in calculating the determinant, the student must determine whether the indices of the expression mean a seventh-order substitution, and whether the substitution sign depends on the number of inversions in it and so on, then remembers and identifies the sign.
The second expression involved eight multipliers. With no repetitions in the second indices, 8 participated twice in the first indices. That is, two elements from row 8 are involved in multiplication. This contradicts the definition of determinant. Hence, the second expression does not participate in the calculation of the 8th order determinant. So is it possible to identify the sign even though it is? If a careful study of the concept of determinant has helped in finding the answer to the first part of the problem given above, it is helpful to know a number of concepts used to illuminate the topic of substitution in the second part, especially how substitution is a subordinate reflection.
References
1. Barton B. The language of mathematics // Springer Science+Business Media. LLC, 2008.
2. Hiehler R., Scholz R.W., Straesser R., Winkelmann B. Didactics of mathematics as a scientific discipline // Kluwer Academic Publishers. New York, 2002.
3. Cowan P. Teaching mathematics a handbook for primary and secondary school teachers // Taylor & Francis e-Library, 2006.
4. Rasulova Z.D. Pedagogical peculiarities of developing socio-perceptive competence in learners // European Journal of Research and Reflection in Educational Sciences. 8:1 (2020). Pp. 30-34.
5. Rasulov T.H., Rasulova Z.D. Organizing educational activities based on interactive methods on mathematics subject // Journal of Global Research in Mathematical Archives. 6:10 (2019). Pp. 43-45.
6. Rashidov A. Development of creative and working with information competences of students in mathematics // European Journal of Research and Reflection in Educational Sciences. 8:3 (2020). Part II. Pp. 10-15.
7. Rasulov T.H. On the finiteness of the discrete spectrum of a 3x3 operator matrix // Methods of Functional Analysis and Topology. 22:1 (2016). P. 48-61.
8. Muminov M.I., Rasulov T.H. On the eigenvalues of a 2x2 block operator matrix // Opuscula Mathematica. 35:3 (2015). P. 369-393.
9. Muminov M.I., Rasulov T.Kh. An eigenvalue multiplicity formula for the Schur complement of a 3x3 block operator matrix // Siberian Math. J. 56:4 (2015). P. 878-895.
10. Muminov M.I., Rasulov T.H. Infiniteness of the number of eigenvalues embedded in the essential spectrum of a 2x2 operator matrix // Eurasian Mathematical Journal. 5:2 (2014). P. 60-77.
11. Rasulov T.Kh. Study of the essential spectrum of a matrix operator // Theoret. and Math. Phys. 164:1 (2010). P. 883-895.
12. Rasulov T.H., Dilmurodov E.B. Eigenvalues and virtual levels of a family of 2x2 operator matrices // Methods of Functional Analysis and Topology. 25:1 (2019). Pp. 273-281.
13. Rasulov T.H., Dilmurodov E.B. Threshold analysis for a family of 2x2 operator matrices // Nanosystems: Physics, Chemestry, Mathematics. 10:6 (2019). P. 616-622.
14. Rasulov T.Kh. On the number of eigenvalues of a matrix operator // Siberian Math. J. 52:2 (2011). P. 316-328.
15. Rasulov T.Kh., Umarova I.O. Spectrum and resolvent of a block operator matrix // Siberian Electronic Mathematical Reports. 11 (2014). P. 334-344.
16. Muminov M.I., Rasulov T.H. On the number of eigenvalues of the family of operator matrices // Nanosystems: Physics, Chemistry, Mathematics. 5:5 (2014). Pp. 619-626.
17. Muminov M., Neidhardt H., Rasulov T. On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case // Journal of Mathematical Physics, 56 (2015), 053507.
18. Muminov M.I., Rasulov T.H. Embedded eigenvalues of an Hamiltonian in bosonic Fock space // Comm. in Mathematical Analysis. 17:1 (2014). P. 1-22.
19. Rasulov T.H. The finiteness of the number of eigenvalues of an Hamiltonian in Fock space // Proceedings of IAM. 5:2 (2016). P. 156-174.
DIALECTICAL THINKING IN THE GERMAN PHILOSOPHY
OF CREATIVITY
1 2 Grigoryeva A.Yu. , Ibragimova K.E.
'Grigoryeva Anastasia Yuryevna - Student; 2Ibragimova Karlygash Eralievna - Master of Education, Senior Lecturer, DEPARTMENT OF THEORY AND PRACTICE OF FOREIGN LANGUAGES, EURASIAN NATIONAL UNIVERSITY NAMED AFTER L.N. GUMILYOV, NUR-SULTAN, REPUBLIC OF KAZAKHSTAN
Abstract: the given article contains information on the attitude of German philosophy towards the understanding of creativity and its connection with the formation of thinking in a child consciousness as a judicious and creative person. In the written article we give broad explanation of such concepts as: thinking, judgment ability, logic, "oneness", dialectics, imagination and its usage in human life. We also scrutinized how dialectical (creative) thinking is considered by German philosophers as crucial point to the formation of an individual personality, and therefore of future generations, and contributes to the development of mankind. In the course of writing of the article, we referred to famous figures of German classical philosophy: Kant I., Hegel G.V.F., Fikhte I.G., and also works byDushinA.V., KojeveA, ChernyakL.S. andAbdildin Zh.M. Keywords: thinking, philosophy, logic, creativity, imagination, reality.
In the course of the formation of philosophy, the most in-depth study of the dialectics of a creative person was considered by German classical philosophy. A single line in the formation of a dialectical interpretation of the phenomenon of a creative personality is manifested in the logical relation of antiquity and German classics.
Creativity, as the highest form of human manifestation, is included in the circle of problems of German classical philosophy, which tried to cognize reality through its multiple phenomena. Systematically approaching the solution of the problems posed, we see the world as one and interconnected.
The "oneness", which was first designated in antiquity, permeates German classical philosophy in various ways. In German classical philosophy, the logic of the development of being was revealed, for which dialectics is of particular significance. Dialectical philosophy has become a holistic understanding of reality. As Kojeve A points out, "dialectics is the proper, true nature of things themselves, and not the "way" of description that is external to them" [1]. It made it possible to consider problems that were previously inaccessible to human cognition, since many elements of thinking and cognition could not be combined within one concept.
German classical philosophy saw this highest development in their own culture. It should be noted that creativity in German classical philosophy was also considered a
