Linear Algebra

Description

The course aims to develop students’ theoretical knowledge and analytical skills in the fundamental concepts and methods of linear algebra. The course covers matrices and operations on matrices, calculation of determinants and their fundamental properties, vectors, vector spaces, and linear spaces. Systems of linear algebraic equations, conditions for the existence of solutions, and various solution methods are examined, along with the fundamental principles of linear transformations. The concepts of eigenvalues and eigenvectors, polynomials and methods for solving polynomial equations, quadratic forms, and their reduction to canonical form are studied. The course also covers fundamental numerical methods and decomposition approaches used in solving linear algebra problems. The course is designed to develop students’ ability to formulate mathematical problems formally, select appropriate solution methods, construct solution algorithms, and evaluate their efficiency.

Course learning outcomes

1. 1

   CLO 1: Define matrices and identify different types of matrices, perform fundamental operations on matrices, explain the main properties of these operations, and transpose matrices. calculate second- and third-order determinants using appropriate methods.

2. 2

   CLO 2: Explain the concepts of minors and cofactors, the expansion of a determinant along a row or column, the inductive definition of higher-order determinants, and methods for calculating them. calculate the determinant of the product of two matrices and find the inverse of a square matrix. perform operations involving orthogonal and block matrices.

3. 3

   CLO 3: Explain the concept of an n-dimensional vector, define linear dependence of a system of vectors, state the necessary and sufficient conditions for linear dependence, and apply methods for calculating the rank of a matrix. define a linear space, basis, dimension, linear transformation, matrix representation of a linear transformation, and operations on linear transformations. explain eigenvalues and eigenvectors of matrices and apply algorithms for finding eigenvalues and corresponding eigenvectors.

4. 4

   CLO 4: Explain systems of linear algebraic equations, including homogeneous and non-homogeneous systems, as well as consistent and inconsistent systems. solve systems of linear algebraic equations using cramer’s rule and gaussian elimination, and solve homogeneous systems of linear equations.

5. 5

   CLO 5: Explain polynomials, arithmetic operations on polynomials, the polynomial division algorithm, bézout’s theorem, and horner’s scheme. apply lagrange and newton methods for determining bounds on the real roots of a polynomial.

6. 6

   CLO 6: Define quadratic forms, express quadratic forms in matrix form, and identify their canonical form. apply the lagrange and jacobi methods to reduce quadratic forms to canonical form.