COURSE GOALS: Acquire knowledge and understanding of the basic concepts of linear algebra. Acquire operational knowledge from methods used to solve problems in linear algebra.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
1. KNOWLEDGE AND UNDERSTANDING
1.5. describe the framework of natural sciences;
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.2. recognize and follow the logic of arguments, evaluate the adequacy of arguments and construct well supported arguments;
2.3. use mathematical methods to solve standard physics problems;
2.6. create a learning environment that encourages active engagement in learning and promotes continuing development of pupils' skills and knowledge;
2.7. plan and design appropriate teaching lessons and learning activities based on curriculum goals and principles of interactive enquirybased teaching;
3. MAKING JUDGMENTS
3.1. develop a critical scientific attitude towards research in general, and in particular by learning to critically evaluate arguments, assumptions, abstract concepts and data;
3.2. develop clear and measurable learning outcomes and objectives in teaching based on curriculum goals;
4. COMMUNICATION SKILLS
4.1. communicate effectively with pupils and colleagues;
4.2. present complex ideas clearly and concisely;
5. LEARNING SKILLS
5.1. search for and use professional literature as well as any other sources of relevant information;
5.2. remain informed of new developments and methods in physics and education;
5.3. develop a personal sense of responsibility for their professional advancement and development;
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
1. Define and understand the concepts described in the course content and apply them in exercises;
2. Identify, discuss and explain the concepts described in the course content;
3. Apply, analyse and link the terms of the course content with problems appearing in further study of physics (e.g. the concepts matrix inverse, determinant, linear operator, eigenvalue, eigenvector, diagonalization of symmetric matrix, Jacobi method);
4. vector spaces, matrices, linearly independent vectors, base of a space, rank of a matrix, homogeneous and nonhomogeneous system of linear equations, elementary transformations, the reduced form, solvability of linear system, the Gauss elimination)
5. To judge and determine which of the methods is best suited for solving the eigenvalue problem for full symmetric matrices or for the inverse of a matrix;
COURSE DESCRIPTION:
Lectures per weeks (15 weeks in total):
1. Linear matrix equations, left and right inverses, nonsingular matrix;
2. Elementary matrices; GaussJordan algorithm for computing the inverse;
3. Major classes of matrices: symmetric, skewsymmetric, orthogonal, permutation matrices; the trace of the matrix, the quadratic form, positive definite matrices;
4. Determinant, definition and the basic properties;
5. BinetCauchy theorem, expanding determinant along the rows and columns;
6. Calculation of the determinant, adjoint and Cramer's rule with examples;
7. Linear operators, isomorphism, coordinatization;
8. Matrix representation of a linear operator, composition of linear operators;
9. Base changes, equivalent and similar matrices; determinant of linear operator;
10. Eigenvalues and eigenvectors of linear operator, characteristic and minimal polynomial;
11. Matrix eigenvalue problem, diagonalizable matrices, matrix polynomial, matrix functions;
12. HamiltonCayley theorem, Jordan form of a matrix;
13. Schur theorem, spectral theorem for normal and symmetric matrices;
14. Diagonalization of symmetric matrix, Jacobi method for symmetric matrix of order two;
15. Jacobi method for symmetric matrix of order n, the algorithm and the basic properties.
REQUIREMENTS FOR STUDENTS:
Students must solve 50% of the written exams (two times in the semester). Each exam consists of two parts: examples and theory, and is evaluated separately. Students must have lecture attendance of at least 50% and should collect at least 48 out of 100 points (which also include home works).
(Second semester, 30 hours of lectures plus 30 hours of exercises)
GRADING AND ASSESSING THE WORK OF STUDENTS:
Grading and assessing the work of students during the semesters:
* Two written exams
* Home works
Grading at the end of semester:
* final oral exam for those who collected at least 88% points and wish grade A
Contributions to the final grade:
* 10% of the grade is carried by the results of the home works
* 10% of the grade will be based on presence (attendance)
* 40% of the grade is carried by the results of the two written exams.
