M2110 Linear Algebra and Geometry II

Faculty of Science
Spring 2025
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: z (credit).
In-person direct teaching
Teacher(s)
prof. RNDr. Jan Paseka, CSc. (lecturer)
doc. RNDr. Martin Čadek, CSc. (seminar tutor)
Mgr. Mária Šimková (seminar tutor)
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
Prerequisites
M1110 Linear Algebra I || M1111 Linear Algebra I ||( FI:MB003 Linear Algebra and Geometry I )
Knowledege of basic notion of linear algebra is supposed.
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
there are 7 fields of study the course is directly associated with, display
Course objectives
The aim of this second course in linear algebra is to introduce other basic notions of linear algebra. Passing the course the students *will know affine spaces, bilinear and quadratic forms, eingenvalues and eigenvectors of linear operators, *they will be able to solve problems concerning the spaces with scalar product and properties of orthogonal and selfadjoint operators and *to find the Jordan canonical form.
Learning outcomes
Passing the course the students *will know affine spaces, bilinear and quadratic forms, eingenvalues and eigenvectors of linear operators, *they will be able to solve problems concerning the spaces with scalar product and properties of orthogonal and selfadjoint operators and *to find the Jordan canonical form.
Syllabus
  • Affine geometry: affine spaces and subspaces, affine geometry and affine mappings. Linear forms: dual space, dual basis, dual homomorphism. Bilinear and quadratic forms: definition, matrix with respect to given basis, diagonalization, signature. Euklidean geometry: orthogonal projection, distance and deviation of affine subspaces. Linear operators: invariant subspaces, eigenvalues and eigen vectors, charakteristic polynomial, algebraic and geometric multiplicity of eigenvalues, conditions for diagonalization. Linear models. Ortogonal and unitar operators: definition and basic properties, eigenvalues, geometric meaning. Self adjoint operators: adjoint operator, symmetric and hermitian matrices, spectral decomposition. Jordan canonical form: nilpotent endomorphisms, root subspaces, computations.
Literature
  • PAVOL, Zlatoš. Lineárna algebra a geometria (Linear algebra and geometry). Bratislava: Albert Marenčin PT, s.r.o., 2011, 741 pp. ISBN 978-80-8114-111-9. info
  • PASEKA, Jan and Pavol ZLATOŠ. Lineární algebra a geometrie I. Elportál. Brno: Masarykova univerzita, 2010. ISSN 1802-128X. URL info
  • Zlatoš P.: Lineárna algebra a geometria, připravovaná skripta MFF Univerzity Komenského v Bratislavě, elektronicky dostupné na http://www.math.muni.cz/pub/math/people/Paseka/lectures/LA/
  • Slovák, Jan. Lineární algebra. Učební texty. Brno:~Masarykova univerzita, 1998. 138. elektronicky dostupné na http://www.math.muni.cz/~slovak.
Teaching methods
Lectures and exercises (tutorials).
Assessment methods
Exam: written and oral. Requirements for the exam: to obtain 50% of points from 6 tests written during semester. Requirements: to manage the theory from the lecture, to be able to solve the problems similar to those from exercises
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Information on completion of the course: ukončení zápočtem možné pouze rozhodnutím učitele
The course is taught annually.
The course is taught: every week.
Credit evaluation note: 2 kr. zápočet.
Listed among pre-requisites of other courses
Teacher's information
http://www.math.muni.cz/~cadek
The exam consists of three parts: 1. Semester-long component: You need to score at least 50% of the points in 6 short written tests. 2. Written exam during the exam period: The written exam consists of a numerical and a theoretical part. Students who pass both parts of the written exam proceed to the oral exam. 3. Oral exam: During the oral exam, you will be required to demonstrate understanding of the topics covered and the ability to illustrate the concepts and theorems with examples. Additional notes: The exam is designed to test your understanding of the material, not your ability to memorize definitions. You are encouraged to ask questions during lectures and tutorials if you do not understand something. There are many resources available to help you prepare for the exam, including the textbook, lecture notes, and online resources.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2000, Spring 2001, Spring 2002, Spring 2003, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024.
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