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Lecturer(s)
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Plešinger Martin, doc. Ing. Ph.D.
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Course content
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1. Fields. Rings, divisibility, zero divisors. Integral domain. Finite rings, fields. Divisibility in domain of integers. Divisibility of linear combination, transitivity of divisibility. (Im)proper divisors. Divisors of unity (invertible elements), associated elements. 2. Common divisors, coprime integers. Euler function, Euler-Fermat theorem. 3. Greatest common divisor (gcd). (Extended) Euclidean algorithm (gcd(a, b) as linear combination of integers a, b). Common multiples, least common multiple [lcm]. 4. Primes and composed integers. Sieve of Eratosthenes. Prime number theorem (number of primes <= x). Factoring integer as product of prime powers, (gcd) a [lcm] from prime factorization. TEST. 5. Polynomials of one variable/indeterminate. Operations, properties. Zeros (roots) of polynomials. Standard forms. Relations between coefficient ring R and polynomial ring R(x). 6. Complex numbers, nth root of 1. Fundamental theorem of algebra. 7. Factorization of polynomial as product of root factors. Multiplicity of roots. Roots of a polynomial and of its derivative. Irreducible polynomials over coefficient field. 8. Division of a polynomial by a binomial (x - c). Horner algorithms. Interpretation of intermediate results (Bezout theorem). Division of a polynomial by a polynomial. Divisibility of polynomials. Common divisors. Euclidean algorithm for polynomials. Modular arithmetic of polynomials. TEST. 9. Algebraic equations with one unknown. Properties of roots of algebraic equations. Algebraic equations of degree <= 4. Solvability in radicals. 10. Special algebraic equations, binomial. Integer, rational roots of algebraic equations with integer, rational coefficients. 11. Some criteria of divisibility (with base b) based on modular arithmetic and Euler-Fermat theorem. 12. Linear mapping: kernel and image, matrix. Geometry of linear objects. Eigenvalues and eigenvectors of a matrix. Geometric interpretation. Canonical form of a matrix. TEST. 13. Linear, bilinear and quadratic forms (matrix approach). Signature of a quadratic form. Applications: classification of quadrics, stationary points, optimization. 14. Time reserve. Continuously - applications of symbolic manipulators (CAS). Seminars accompany lectures in accordance with instructions of lecturer. ----- The range of lessons for extra-mural study: 24 hours/semester
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Home preparation for classes
- 42 hours per semester
- Preparation for credit
- 28 hours per semester
- Class attendance
- 84 hours per semester
- Preparation for exam
- 56 hours per semester
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Learning outcomes
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This subject encloses the linear algebra lecture by focusing on the singular value decomposition. Then we switch to the general algebra topics: * modular arithmetics, commutative rings, integral domains; * introduction to group theory; * fields.
Theory, algorithms and applications of number theory, esp. of modular arithmetic. Theory, algorithms and applications of polynomial algebra. Matrix algebra, geometric interpretations. Applications in geometry and optimization.
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Prerequisites
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Secondary school maths, Algebra and geometry 1
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Assessment methods and criteria
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Oral exam, Written exam
Write, send, and present semester work. Passing of tests, written and oral exam.
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Recommended literature
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Bican, L. Algebra (pro učitelské studium). Praha, Academia, 2001. ISBN 80-200-0860-8.
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Cameron, P. J. Introduction to Algebra. Oxford University Press, 2008.
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Durbin, J. R. Modern Algebra: An Introduction. Willey, 2008.
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Stanovský, D. Základy algebry. Matfyzpress, 2010.
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