Course: Linear algebra and discrete mathematics

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Course title Linear algebra and discrete mathematics
Course code NTI/LADM
Organizational form of instruction Lecture + Lesson
Level of course Bachelor
Year of study not specified
Semester Winter
Number of ECTS credits 7
Language of instruction Czech
Status of course Compulsory
Form of instruction Face-to-face
Work placements Course does not contain work placement
Recommended optional programme components None
Lecturer(s)
  • Stebel Jan, doc. Mgr. Ph.D.
  • Rálek Petr, Ing. Ph.D.
Course content
Lectures: 1. Vectors in R^n, linear combination, scalar product, norm, law of cosines, Schwarz inequality. 2. Systems of linear algebraic equations. Row, column and matrix representation. 3. Matrices. Basic matrix operations and types, inverse matrix. 4. Gauss and Gauss-Jordan elimination, LU decomposition. 5. Linear vector space, linear dependence, span, basis and dimension of linear space. 6. Nullspace and range of matrix, matrix rank. Frobenius theorem. General solutions of system of linear equations. 7. Orthogonal subspaces, projection. Fundamental theorem of linear algebra. 8. System of normal equations. Least squares method. 9. Orthogonal matrix, Gram-Schmidt process, QR decomposition. 10. Permutation, determinant and its calculation, expansion of determinant by row or column. Cramer's rule. 11. Eigenvalues and eigenvectors of square matrices. Diagonalization, spectral theorem. Jordan decomposition. 12. Linear transformation, its matrix associated to bases. Null-space and image of linear transformation. 13. Coordinates, transition matrix, change of matrix of linear map due to change of basis. Singular value decomposition. 14. Graph, distance in graph, shortest path. Dijkstra's algorithm. Tree, spanning tree, Kruskal's and Jarník's algorithms. Topics of tutorials correspond to topics of lectures.

Learning activities and teaching methods
Monological explanation (lecture, presentation,briefing)
  • Contacts hours - 84 hours per semester
  • Preparation for exam - 51 hours per semester
  • Preparation for credit - 38 hours per semester
  • Preparation for formative assessments - 36 hours per semester
Learning outcomes
The course covers basic parts of linear algebra and discrete mathematics necessary for deeper understanding of principles of natural sciences. Theoretical presentation of the lectures will be followed by solving particular problems at tutorials.
The student will gain basic knowledge of linear algebra and discrete mathematics. Tosome extent they will learn abstract sensing, formulation and solution of real world problems leading to systems of linear equations, matrix or graph problems.
Prerequisites
Unspecified

Assessment methods and criteria
Combined examination

Requirements for getting a credit are activity at the tutorials and successful passing of the tests. Examination is written and oral form.
Recommended literature
  • Bican L. Lineární algebra a geometrie. Academia, 2009.
  • J. Matoušek, J. Nešetřil. Kapitoly z diskrétní matematiky. Karolinum, 2009. ISBN 80-246-0084-.
  • Strang, G. Introduction to Linear Algebra. Wellesley-Cambridge Press, 2003. ISBN 0-9614088-2-0.


Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester
Faculty: Faculty of Mechatronics, Informatics and Interdisciplinary Studies Study plan (Version): Applied Sciences in Engineering (2019) Category: Special and interdisciplinary fields 1 Recommended year of study:1, Recommended semester: Winter