| Course Unit Code | 470-4201/01 |
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| Number of ECTS Credits Allocated | 4 ECTS credits |
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| Type of Course Unit * | Optional |
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| Level of Course Unit * | Second Cycle |
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| Year of Study * | First Year |
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| Semester when the Course Unit is delivered | Summer Semester |
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| Mode of Delivery | Face-to-face |
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| Language of Instruction | Czech |
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| Prerequisites and Co-Requisites | Course succeeds to compulsory courses of previous semester |
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| Name of Lecturer(s) | Personal ID | Name |
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| DOS35 | prof. RNDr. Zdeněk Dostál, DSc. |
| VLA04 | Ing. Oldřich Vlach, Ph.D. |
| Summary |
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| Vector space, orthogonality, special bases (hierarchical, Fourier, wavelets), linear mapping, bilinear an quadratic forms, matrix decompositions (spectral, Schur, SVD), Markov's precesses, Page Rank vector, linear algebra of huge matrices, low rank approximation of large matrices, quadratic programming, SVM, tensors. Applications in information technology. |
| Learning Outcomes of the Course Unit |
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| A sudent will get basic knowledge of linear and multilinear algebra and their applications in modern information technology. |
| Course Contents |
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• An introduction to matrix decompositions with motivation and applications
• Spectral decomposition of a symmetric matrix
• Applications of the spectral decomposition: matrix functions, convergence of iterative methods, extremal properties of the eigenvalues
• QR decomposition – rank of the matrix, atable solution of linear systems, reflection
• SVD – low rank approximations of a matrix, image deblurring, image compression
• Approximate decompositions of large matrices and related linear algebra
• Tensor approximations – Kronecker product, tensors, tensor SVD, tensor train, image debluring
• Variational principle and least squares
• Total least squares
• Minimization of a quadratic function with equality constraints – KKT, duality, basic algorithms, SVM,
• Analytic geometry with matrix decompositions
• Inverse problems – Tichonov regularization, applications
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| Recommended or Required Reading |
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| Required Reading: |
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N. Halko, P. G. Martinsson, J. A. Tropp: Finding Structure with Randomness:
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions,
SIAM REVIEW, Vol. 53, No. 2, (2011)217–288
Matrix Analysis for Scientists and Engineers
by Alan J. Laub, SIAM, Philadelphia
Alan J. Laub, Matrix Analysis for Scientists and Engineers, SIAM, Philadelphia, 2005
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Zdenek Dostál, Lineární algebra, VŠB Ostrava 2000
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| Recommended Reading: |
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Tamara G. Kolda, Brett W. Bader. Tensor Decompositions and Applications, SIAM Review, Vol. 51, No. 3, (2009)455–500
Carl D. Meyer, Matrix analysis and applied linear algebra, SIAM, Philadelphia, 2000
Dianne P. O'Leary, Scientific Computing with Case Studies, SIAM, Philadelphia 2009 |
Milan Hladík, Lineární algebra(nejen)pro informatiky, MFF UK 2019 (pdf na https://kam.mff.cuni.cz/~hladik/LA/text_la_upd.pdf)
Luboš Motl, Miloš Zahradník : Pěstujeme lineární algebru. MFF UK 2011 (http://matematika.cuni.cz/zahradnik-pla.html) |
| Planned learning activities and teaching methods |
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| Lectures, Tutorials |
| Assesment methods and criteria |
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| Task Title | Task Type | Maximum Number of Points
(Act. for Subtasks) | Minimum Number of Points for Task Passing |
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| Credit and Examination | Credit and Examination | 100 (100) | 51 |
| Credit | Credit | 30 | 15 |
| Examination | Examination | 70 | 21 |