F2182 Linear and multilinear algebra

Faculty of Science
Spring 2025
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. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: Mgr. Pavla Musilová, Ph.D.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
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
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 the 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 features and 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.
Learning outcomes
Absolving the discipline student obtains following knowledge 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. Self-adjoint 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
    required literature
  • MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika II pro porozumění i praxi (Mathematics II for understanding and praxis). první. Brno: VUTIUM (Vysoké učení technické v Brně), 2012, 697 pp. ISBN 978-80-214-4071-5. info
    recommended 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.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2000, Spring 2001, Spring 2002, Spring 2003, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024.
  • Enrolment Statistics (recent)
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