M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2011
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 12:00–13:50 M1,01017
  • Timetable of Seminar Groups:
M3130/01: Tue 14:00–15:50 M4,01024, L. Vokřínek
M3130/02: Wed 16:00–17:50 M5,01013, L. Vokřínek
Prerequisites
M2110 Linear Algebra and Geometry II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of quadratic and bilinear forms.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Abstract
The third from the series of lectures on linear algebra and geometry is devoted to the following three topics: quadrics and their classification, multilinear algebra and connection between polynomial matrices and the Jordan canonical form.

These topics find applications in differential geometry and linear differential equations.

At the end of this course students should be able to:
*understand the connection between bilinear forms and the geometry of quadrics
*derive geometric properties of quadrics
*compute with tensors both in coordinates and without them
*utilize a different method of finding the Jordan canonical form of a matrix
Key topics
  • Affine and projective spaces: definitions, subspaces, homomorphisms, projective closure, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Selected applications: spectral decomposition, Moore-Penrose pseudoinverse, Markov chains
  • Multilinear algebra: dual space, tensor product, exterior and symmetric products, tensor coordinates, functor Hom and its relation to the tensor product.
  • Integer matrices: equivalence, Smith normal form, classification of finitely generated commutative groups
  • Polynomial matrices: equivalence, Smith normal form, connection with characteristic and minimal polynomial, Jordan canonical form.
Study resources and literature
  • Čadek M: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~cadek
  • Slovák J.: Lineární algebra, elektronický učební text PřF MU Brno, www.math.muni.cz/~slovak
  • Kostrikin A., Manin Yu.: Linear algebra and geometry, Gordon and Breach Science Publishers, 1997
  • LANG, Serge. Linear Algebra. Third Edition. New York: Springer-Verlag, 1987, 296 pp. ISBN 0-387-96412-6. info
Approaches, practices, and methods used in teaching
Lectures and tutorials.
Method of verifying learning outcomes and course completion requirements
Over the semester there will be two written tests at the tutorials. It is required to obtain in total at least half of the points.
The exam consists of a written and an oral part.
Language of instruction
Czech
Follow-Up Courses
Teacher's information
http://www.math.muni.cz/~cadek
Ke zkoušce je nutný zápočet ze cvičení. Zkouška je písemná a ústní. Písemná zkouška má početní a teoretickou část. Při ústní zkoušce se bude ověřovat porozumění přednesené látce a schopnost ji demonstrovat na jednoduchých příkladech. Nutná je znalost základních pojmů lineární algebry z předchozích semestrů: vektorový prostor, báze, souřadnice, lineární zobrazení, matice lineárního zobrazení, matice přechodu, vlastní čísla a vektory, bilineární a kvadratické formy.
Na cvičeních se budou psát dvě písemky. Zápočet bude udělen, pokud student za tyto písemky obdrží aspoň polovinu z udělených bodů. Studenti, kteří dosáhnou horšího výsledku, budou psát náhradni zápočtovou písemku. Na zápočet musí získat aspoň polovinu bodů.
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024, Autumn 2025, Autumn 2026.
  • Enrolment Statistics (Autumn 2011, recent)
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