PřF:M3130 Linear Algebra III - Course Information

M3130 Linear Algebra and Geometry III

Extent and Intensity

2/2/0. 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

Tue 12:00–13:50 M5,01013

• Timetable of Seminar Groups:

M3130/01: Wed 14:00–15:50 M2,01021, L. Vokřínek

Prerequisites

M2110 Linear Algebra II
Knowledge of basic notions of linear algebra including eigenvalues and eigenvectors, knowledge of bilinear and quadratic 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

• Mathematics (programme PřF, B-MA)
• Statistics and Data Analysis (programme PřF, B-MA)

Course objectives

The third from the series of lectures on linear algebra and geometry is devoted to the following three topics:
- polyhedra and optimalization of linear functions on polyhedra,
- multilinear algebra and tensors,
- integer and polynomial matrices and their connection with Jordan canonical form.
These topics find applications in differential geometry and linear differential equations.

Learning outcomes

At the end of this course students will be able to:
- understand the structure of polyhedra and solve the linear programming problem via the simplex method;
- compute with tensors both in coordinates and without them;
- find the Smith normal form of a matrix and interpret it, in particular, derive from it the Jordan canonical form

Syllabus

• Affine and projective spaces: definitions, subspaces, homomorphisms, complexification.
• Polyhedral cones and polyhedra: various definitions and their comparison, Farkas lemma, faces of polyhedra, linear programming problem, duality in linear programming, simplex method
• Multilinear algebra: dual space, tensor product, external and symmetric products, coordinates of tensors, functor Hom and its relation to the tensor product.
• Integer and polynomial matrices: Smith normal form, connection with presentations of commutative groups, classification of finitely generated commutative groups, connection with characteristic and minimal polynomial and with Jordan canonical form.

Literature

• Čadek M, Vokřínek L: Lineární algebra a geometrie III, elektronický učební text PřF MU Brno, www.math.muni.cz/~koren
• 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 tests at the tutorials. It is required to obtain in total more than half of the points from these tests. In addition, the maximum allowed number of absences at the tutorials is set to 3 (in the case of the distance education, the presence will be supplemented by working out homeworks).
The final exam consists of a written and an oral part. It is required to obtain at least 50% of points from the written part.

Language of instruction

Czech

Follow-Up Courses

• M4190 Differential Geometry of Curves and Surfaces

Further comments (probably available only in Czech)

Study Materials
The course is taught annually.

Teacher's information

http://www.math.muni.cz/~koren

• Enrolment Statistics (autumn 2021, recent)
  
• Permalink: https://is.muni.cz/course/sci/autumn2021/M3130