Lecturer(s)
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Stebel Jan, doc. Mgr. Ph.D.
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Rálek Petr, Ing. Ph.D.
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Course content
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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.
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Learning activities and teaching methods
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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
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Learning outcomes
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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.
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Prerequisites
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Unspecified
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Assessment methods and criteria
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Combined examination
Requirements for getting a credit are activity at the tutorials and successful passing of the tests. Examination is written and oral form.
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Recommended literature
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Bican L. Lineární algebra a geometrie. Academia, 2009.
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J. Matoušek, J. Nešetřil. Kapitoly z diskrétní matematiky. Karolinum, 2009. ISBN 80-246-0084-.
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Strang, G. Introduction to Linear Algebra. Wellesley-Cambridge Press, 2003. ISBN 0-9614088-2-0.
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