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Lecturer(s)
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Botur Michal, doc. Mgr. Ph.D.
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Cenker Václav, Mgr.
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Course content
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2. Matrices: Operations with matrices, vector space of matrices, ring of square matrices. 3. Determinants: Definition, calculation of determinants. 4. Vector spaces: Subspace, subspace generated by a set, basis, dimension. 5. Systems of equations: Homogeneous and nonhomogeneous systems and their solutions, the Frobenius theorem, Gauss elimination, the Cramer rule. 6. Homomorphisms and isomorphisms of vector spaces: Arithmetical vector spaces and their importance for description of vector spaces, coordinates of vectors according to a given basis, transformation of coordinates as consequense of change of basis, matrix of transformation, matrix of endomorphism. 8. Ring of polynomials and its properties: Functional and algebraic definition from the structural point of view. 9. Divisibility of polynomials over a general field: Properties of the structure (T x , +, ) concerning divisibility. 10. Properties of polynomial roots: Root of a polynomial, multiplicity of the root, the Bezout theorem, the Horner scheme, derivative of a polynomial and its application, the Basic Theorem of Algebra, decomposition of polynomials to product of irreducible ones over Q, R and C, the Viéte theorem, methods of root solving of polynomials. 11. Algebraic solvability of algebraic equations: Extension of fields using radicals, algebraic solvability of algebraic equations with respect to degree. 12. Quadratics and bilinear forms. 13. Numerical methods of solving algebraic equations: Essence of numerical methods, basic methods of separation and aproximation of real roots.
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Learning activities and teaching methods
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Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
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Learning outcomes
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Understand bases of linear algebra, to master solving the typical tasks. Understand bases of theory of polynomials, to master solving the typical tasks.
3. Aplication Students obtain ability to apply knowledge of linear algebra for solving of particular mathematical problems and knowledge of theory of polynomials and algebraic equations for solving of particular equations
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Prerequisites
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unspecified
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Assessment methods and criteria
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Oral exam, Written exam
Exam: the student has to understand the subject and be able to prove the principal results.
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Recommended literature
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Bican, L. (2000). Lineární algebra a geometrie. Praha, Academia.
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Bican L. (1979). Lineární algebra. SNTL Praha.
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Blažek J. (1985). Algebra a teoretická aritmetika I. SPN Praha.
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Emanovský P. (2002). Algebra 2, 3 (pro distanční studium). VUP Olomouc.
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Emanovský P. (1998). Cvičení z algebry (polynomy, algebraické rovnice). VUP Olomouc.
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Hort D., Rachůnek J. (2003). Algebra I. UP Olomouc.
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Katriňák T. (1985). Algebra a teoretická aritmetika (1). Alfa Bratislava.
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Kořínek V. (1956). Základy algebry. NČSAV Praha.
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Waerden, L. (1971). Algebra I. Springer-Verlag Berlin, Heidelberg, New York.
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