PřF:F2182 Linear and multilinear algebra - Course Information
F2182 Linear and multilinear algebra
Faculty of ScienceSpring 2011 - only for the accreditation
- Extent and Intensity
- 3/1/0. 3 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Pavla Musilová, Ph.D. (seminar tutor) - Guaranteed by
- prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: Mgr. Michael Krbek, Ph.D. - Prerequisites
- Fundamental knowledge of algebraic structures, fundamentals of matrix theory and their application for solving systems of linear equations.
- Course Enrolment Limitations
- The course is also offered to the students of the fields other than those the course is directly associated with.
- fields of study / plans the course is directly associated with
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, B-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Course objectives
- The discipline is a part of the fundamental course of linear and multilinear algebra for students of physics.
Linear and multilinear algebra is one of most important mathematical tools, practically for all physical theories. Thus, the main goal of the discipline is to give to students a satisfactorily deep understanding of the concept of linear mapping and its fundamental properties. Only by such a way it is possible to ensure a good orientation of students in problems of vector and tensor algebra, the mathematically correct use of tensor calculus and the understanding of the most important featuresand properties of vector and tensor physical quantities. The discipline also gives a thorough preparation of algebraic tools for the theory of integration of forms on euclidean spaces and differential manifolds.
Absolving the discipline student obtains following kowledge and skills:
* Deep understanding of the concept of linearity and linear mapping of vector spaces of general dimensions.
* Understanding of algebraic formulation of geometrical problems (general projection, etc.)
* Practical manipulations with linear mappings with the use of matrix calculus.
* Practical solution of eigenvalue problem for a linear operator, diagonalization.
* Understanding of the concept of a multilinear mapping and its applications in physics (tensor quantities).
* Practical calculus with tensors and tensor operations. - Syllabus
- 1. Linear mappings of finite-dimensional vector spaces: representation of a vector in bases, subspaces.
- 2. Scalar product, orthogonalization, orthogonal projection.
- 3. Linear operators in vector spaces and their representation in bases.
- 4. Eigenvalues and eigenvectors, diagonal representation.
- 5. Unitary linear operators. Selfadjoint linear operators.
- 6. Spectral representation.
- 7. Jordan normal form: Polynomial matrices and Jordan normal form.
- 8. Jordan normal form: JNF and invariant subspaces.
- 9. Tensor algebra: dual space, dual basis. Tensor product of vector spaces.
- 10. Tensors as linear operators, representation in bases, operations.
- 11. Algebraic structure of tensor spaces.
- 12. Exterior algebra.
- 13. Induced mappings of tensor spaces.
- 14. Physical applications-cartesian tensors.
- Literature
- MUSILOVÁ, Jana and Demeter KRUPKA. Lineární a multilineární algebra. 1. vyd. Praha: Státní pedagogické nakladatelství, 1989, 281 s. info
- SLOVÁK, Jan. Lineární algebra. Učební texty. Brno: Masarykova univerzita, 1998, 138 pp. elektronicky dostupné na www.math.muni.cz/~slovak. ISBN nemá. info
- Teaching methods
- Lectures: theoretical explanation with practical examples
Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems, homeworks, tests - Assessment methods
- Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. - Language of instruction
- Czech
- Follow-Up Courses
- Further comments (probably available only in Czech)
- The course is taught annually.
The course is taught: every week.
- Enrolment Statistics (Spring 2011 - only for the accreditation, recent)
- Permalink: https://is.muni.cz/course/sci/spring2011-onlyfortheaccreditation/F2182