Linear algebraTopics

Within the subject, the concept of matrix, the main types of matrices and the basic algebraic operations performed on them are studied. the rules of adding, subtracting, multiplying and multiplying matrices are reviewed. the concept of the determinant of a quadratic matrix, the main properties of the determinant and the calculation principles are explained. methods of calculating determinants of the second and third order and the application of determinants in solving problems related to matrices are also investigated.

In the framework of the subject, the concepts of minor and algebraic complement, their role in the calculation of determinants are studied. the principle of separation of the determinant on selected row or column elements is considered. the methods used in the calculation of higher-order determinants and the essence of laplace's theorem are explained. also, the simplification of the calculation process by choosing the appropriate row or column and the application of laplace's theorem to the calculation of various determinants are investigated.

In the framework of the topic, the concept of an inverse matrix, the conditions for the existence of an inverse matrix and methods of finding it are studied. the concept of the rank of a matrix, the principles of determining the rank, and the calculation of the rank of diagonally given matrices are reviewed. methods of finding the inverse matrix through determinant and elementary transformations are explained. also, the concept of specific matrix and the main properties of such matrices are investigated.

Within the subject, concepts of orthogonal and block matrix, their main characteristics and principles of application are studied. the structure of orthogonal matrices, their main properties and algebraic operations performed on them are considered. the division of matrices into blocks, the structure of block matrices and the rules for performing addition, multiplication and other basic operations on them are explained. also, the existence of inverses of block matrices and methods of finding the inverse block matrix are investigated.

In the framework of the topic, the concepts of n-dimensional vector and n-dimensional vector space are studied. linear combination of vectors, linear dependence and necessary and sufficient conditions of linear dependence are considered. the concept of rank of a matrix and its relationship with linearly independent rows and columns of a matrix are explained. the concept of k-order minor, determining the rank of a matrix based on its non-zero minors, and the basic methods used to calculate the rank of a matrix are investigated.

Within the framework of the topic, the concept of linear space, its main features and structure are studied. concepts of the basis and size of the linear space, the principles of expressing the vectors on the basis and determining their coordinates are considered. the concept of linear transformation, the main properties of linear transformations and their expression through a matrix are explained. also, rules for constructing a linear transformation matrix and performing basic operations such as addition, multiplication, and composition on linear transformations are explored.

Within the subject, the concepts of eigenvalue and eigenvector of a matrix and their main characteristics are studied. algorithms for constructing a characteristic equation, finding eigenvalues ​​and eigenvectors corresponding to them are considered. basic algebraic operations on vectors, addition, subtraction and multiplication of vectors are explained. the concepts of scalar product, vector product and mixed product, their calculation methods, basic properties and geometric meanings are also examined.

In the framework of the topic, the concept of a system of linear algebraic equations and the basic concepts related to these systems are studied. the structure and main properties of systems of homogeneous and non-homogeneous linear algebraic equations are considered. according to the existence of a solution, compensable and non-compensable systems, as well as the classification of compensable systems according to the number of solutions, are explained. the expression of the system of linear algebraic equations in the form of a matrix through the matrix of coefficients, vectors of unknowns and free limits is investigated.

Within the framework of the topic, the main features of systems of non-homogeneous linear algebraic equations and their solution methods are studied. the application conditions of kramer's method, the construction of the main determinant and auxiliary determinants, as well as the principles of determining the solution of the system through these determinants are reviewed. solving the system of inhomogeneous linear equations by the inverse matrix method, the existence condition of the inverse of the matrix of coefficients, and the rules for finding the solution of the system in matrix form are explained. also, application features and solution processes of cramer and inverse matrix methods are comparatively investigated.

The gauss and gauss-jordan methods used in the solution of systems of inhomogeneous linear algebraic equations are studied within the subject. the principles of expressing the system of linear equations in the form of an expanded matrix and bringing it into stepwise form through elementary row transformations are considered. the rules for sequential determination of unknowns using the gaussian method and reducing the matrix to a reduced stepwise form using the gauss-jordan method are explained. also, during the application of these methods, the existence of the solution of the system and the determination of the number of solutions, as well as the comparative characteristics of gauss and gauss-jordan methods are investigated.

In the framework of the topic, the concept, structure and main features of the system of homogeneous linear algebraic equations are studied. the existence of solutions of homogeneous systems, the notions of trivial and non-trivial solutions, as well as the relationship of the set of solutions to the linear space are considered. the principles of bringing the coefficient matrix of the system into stepwise form and constructing a general solution through free variables are explained. also, finding the system of fundamental solutions of the system of homogeneous equations and applying it to practical studies is investigated.

Within the scope of the subject, the concepts of inner product and inner product space, their main properties are studied. definition of the angle between vectors in inner product spaces and the concept of orthogonality are considered. the gram–schmidt process and its application are explained for constructing an orthonormal basis from a system of linearly independent vectors. the essence of qr-decomposition and the expression of matrices in the form of products of orthogonal and triangular matrices are investigated. also, the method of least squares, the problems of approximation of data and functions, as well as the expression of functions on orthogonal systems and the basic principles of fourier series are considered.

Within the subject, the concepts of matrix diagonalization, orthogonal matrices and orthogonal diagonalization are studied. the conditions of matrix diagonalization and the principles of diagonalization through eigenvalues ​​and eigenvectors are reviewed. the concept of quadratic form, the rules for expressing quadratic forms in the form of a matrix and bringing them to a canonical form through lagrangian and jacobian methods are explained. the possibilities of using quadratic forms in solving optimization problems are investigated. also, the concepts, basic properties of hermitian, unitary, and normal matrices and their relation to diagonalization are reviewed.

In the framework of the topic, the concept of polynomials and the basic algebraic operations performed on polynomials are studied. algorithm of residual division of polynomials, concept of root of polynomial and basic principles of determining roots are reviewed. the essence of bezu's theorem and hörner's scheme, their application in the calculation of the value of polynomials and the investigation of roots are explained. also, the application of lagrange's and newton's methods in finding the roots and determining the limits of real roots of polynomials are investigated.

The main methods and matrix decompositions used in the numerical solution of linear algebra problems are studied within the subject. the essence of LU decomposition, the expression of the matrix as a product of lower and upper triangular matrices, and its application in solving systems of linear equations are considered. the power method is explained for the approximate determination of the dominant eigenvalue of a matrix and its corresponding eigenvector. different methods used in solving linear systems are compared in terms of computational efficiency, accuracy, and application characteristics. it also examines the essence of singular value decomposition (svd), the decomposition of matrices into components, and the principles of data reduction and compression using svd.