M3130 Linear Algebra and Geometry III

Faculty of Science
Autumn 2015
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
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Wed 13:00–14:50 M4,01024
  • Timetable of Seminar Groups:
M3130/01: Wed 15:00–16:50 M4,01024, L. Vokřínek
Prerequisites
M2110 Linear Algebra 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
Course objectives
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
Syllabus
  • Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
  • Quadrics in affine and projective spaces: definitions, projective classification, conjugate points, tangent planes, affine classification.
  • Metric properties of quadrics: principal directions, principal planes, metric classification.
  • Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
  • Polynomial matrices: equivalence, canonical forms, connection with characteristic and minimal polynomial, Jordan canonical form.
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
Teaching methods
Lectures and tutorials.
Assessment methods
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
Further comments (probably available only in Czech)
The course is taught annually.
Teacher's information
http://www.math.muni.cz/~cadek
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, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.
  • Enrolment Statistics (Autumn 2015, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2015/M3130