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Lecturer(s)
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Pěchota Jan, RNDr. Ph.D.
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Pekarovič Václav, Mgr.
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Vávrová Miroslava, RNDr.
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Kopal Stanislav, doc. RNDr. Ph.D.
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Štěpánková Magdalena, Bc.
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Oblištilová Aneta Joanna, Bc.
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Hofman Martin, RNDr. Mgr. Ph.D.
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Course content
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1. Complex numbers, fields. Polynomials, rings. Horner scheme, polynomial factorization. 2. Vector space, linear dependence and independence, basis and dimension of vector space, coordinates of vector relative to basis 3. Matrices, determinant of matrix and its basic properties, determinant expansion 4. Gaussian elimination. Fast calculation of determinants. Vector spaces associated with matrix. Rank of matrix, calculation of rank using determinants 5. Matrix inverse, Gauss-Jordan elimination, calculation of matrix inverse using determinants 6. Linear transformation, kernel and image and their dimensions, matrix of linear transformation and its properties. Fundamental Theorem of Linear Algebra. 7. Inverse linear transformation, linear transformation composition and its matrix, vector space isomorphism, change of basis and change-of-basis matrix 8. Systems of linear equations, homogeneous and non-homogeneous systems of equations, linear systems with invertible coefficient matrix, Cramer's rule 9. Eigenvalues and eigenvectors of matrix, generalized eigenvectors. Similarity of matrices. Jordan normal form of matrix. Matrix functions 10. Metric, norm, inner product and their properties. Euclidean and unitary spaces. Orthogonal and orthonormal basis for a vector space 11. Gram-Schmidt process, orthogonal projection onto subspace. QR decomposition of a matrix. 12. Linear least squares regression. Linear forms. Multilinear forms. Quadratic forms and real valued symmetric matrices. Definiteness of a matrix. 13. Inertia of quadratic form, Sylvester's law of inertia for quadratic forms. Quadratic forms and optimization.
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Learning activities and teaching methods
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Interactive lecture, Collaborative instruction
- Preparation for an examination (30-60)
- 54 hours per semester
- Contact hours
- 65 hours per semester
- Preparation for formative assessments (2-20)
- 12 hours per semester
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| prerequisite |
|---|
| Knowledge |
|---|
| to explain the concept of a vector |
| to define the concept of a function |
| to identify equations of basic geometric configurations |
| Skills |
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| to use basics of analytic geometry |
| to solve elementary systems of equations |
| Competences |
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| N/A |
| N/A |
| learning outcomes |
|---|
| Knowledge |
|---|
| to explain the concept of a vector and matrix |
| to describe the concept of a vector space |
| to describe the concept of a linear transformation |
| to characterize eigenvalues and eigenvectors of a matrix |
| Skills |
|---|
| to find roots of polynomials in one variable |
| to calculate determinant of a matrix, matrix inverse and rank of matrix |
| to solve systems of linear algebraic equations |
| to find eigenvalues and eigenvectors of a matrix |
| to use the least squares method |
| Competences |
|---|
| N/A |
| N/A |
| teaching methods |
|---|
| Knowledge |
|---|
| Lecture |
| Skills |
|---|
| Practicum |
| Competences |
|---|
| Lecture |
| Practicum |
| assessment methods |
|---|
| Knowledge |
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| Oral exam |
| Written exam |
| Skills |
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| Test |
| Written exam |
| Competences |
|---|
| Oral exam |
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Recommended literature
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Anton, H.; Rorres, Ch. Elementary Linear Algebra: Applications Version. Wiley, 2013. ISBN 978-1118434413.
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Axler, Sheldon. Linear Algebra Done Right. Springer International Publishing, 2015. ISBN 978-3-319-11079-0.
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Bečvář, Jindřich. Lineární algebra. MatfyzPress, 2020. ISBN 978-80-7378-378-5.
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Bican, Ladislav. Linární algebra a geometrie. Academia, 2009. ISBN 978-80-200-1707-9.
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Demlová, Marie; Nagy, Jozef. Algebra. 2. vyd. Praha : SNTL, 1985.
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Havel, Václav; Holenda, Jiří. Lineární algebra. 1. vyd. Praha : SNTL, 1984.
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Hladík, Milan. Lineární algebra (nejen) pro informatiky. MatfyzPress, 2019. ISBN 9788073783921.
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Motl, Luboš. Pěstujeme lineární algebru. 2. vyd. Praha : Univerzita Karlova, 1999. ISBN 80-7184-815-8.
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Motl, Luboš; Zahradník, Miloš. Pěstujeme lineární algebru. Univerzita Karlova, 2002. ISBN 80-246-0421-3.
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Olver, Peter J.; Shakiban, Chehrzad. Applied Linear Algebra. Springer International Publishing AG, part of Springer Nature, 2018. ISBN 978-3-319-91040-6.
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Tesková, Libuše. Lineární algebra. 1. vyd. Plzeň : Západočeská univerzita, 2001. ISBN 80-7082-797-1.
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Tesková, Libuše. Sbírka příkladů z lineární algebry. 5. vyd. Plzeň : Západočeská univerzita, 2003. ISBN 80-7043-263-2.
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