M2110B 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).
Taught in person.
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
  • 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
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. You need to score a total of 12 points out of 22. 3. Oral exam: Students who pass both parts of the written exam proceed to the 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. Exam requirements: Mastery of the material covered in lectures and tutorials. In the case of distance learning, the material is available on the course website throughout the semester. You will be asked about definitions, theorems, examples, and proofs. Emphasis is placed on understanding. It is not enough to know the definitions and theorems; you need to be able to provide examples of the defined concepts and the main theorems. You are also required to be able to perform simple proofs. Additional notes: The instructor is happy to answer your 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 materials.
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.
Teacher's information
http://www.math.muni.cz/~cadek
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