Course detail

Linear Algebra

FSI-SLAAcad. year: 2020/2021

The course deals with following topics: Sets: mappings of sets, relations on a set.
Algebraic operations: groups, vector spaces, matrices and operations on matrices.
Fundamentals of linear algebra: determinants, matrices in step form and rank of a matrix, systems of linear equations. Euclidean spaces: scalar product of vectors, eigenvalues and eigenvectors. Fundamentals of analytic geometry: linear concepts, conics, quadrics.

Guarantor

doc. RNDr. Miroslav Kureš, Ph.D.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

Students will be made familiar with algebraic operations,linear algebra, vector and Euclidean spaces, and analytic geometry. They will be able to work with matrix operations, solve systems of linear equations and apply the methods of linear algebra to analytic geometry and engineering tasks. When completing the course, the students will be prepared for further study of mathematical and technical disciplines.

Prerequisites

Students are expected to have basic knowledge of secondary school mathematics.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Literature

Thomas, G. B., Finney, R.L.: Calculus and Analytic Geometry, Addison Wesley 2003.
Howard, A. A.: Elementary Linear Algebra, Wiley 2002.
Rektorys, K. a spol.: Přehled užité matematiky I., II., Prometheus 1995.
Searle, S. R.: Matrix Algebra Useful for Statistics, Wiley 1982.
Karásek, J., Skula, L.: Algebra a geometrie, Cerm 2002.
Nedoma, J.: Matematika I., Cerm 2001.
Nedoma, J.: Matematika I., část první: Algebra a geometrie, PC-DIR 1998.
Horák, P., Janyška, J.: Analytická geometrie, Masarykova univerzita 1997.
Janyška, J., Sekaninová, A.: Analytická teorie kuželoseček a kvadrik, Masarykova univerzita 1996.
Mezník, I., Karásek, J., Miklíček, J.: Matematika I. pro strojní fakulty, SNTL 1992.
Horák, P.: Algebra a teoretická aritmetika, Masarykova univerzita 1991.
Procházka, L. a spol.: Algebra, Academia 1990.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Course-unit credit requirements: Active attendance at the seminars, at least 50% of points in written tests. There is one alternative date to correct these tests.
Form of examinations: The exam is written and has two parts.
The exercises part takes 100 minutes and 6 exercises are given to solve.
The theoretical part takes 20 minutes and 6 questions are asked.
At least 50% of the correct results must be obtained from each part. If less is met in one of the parts, then the classification is F (failed).
Exercises are evaluated by 3 points, questions by 1 point.
If 50% of each part is met, the total classification is given by the sum.
A (excellent): 22 - 24 points
B (very good): 20 - 21 points
C (good): 17 - 19 points
D (satisfactory): 15 - 16 points
E (enough): 12 - 14 points
F (failed): 0 - 11 points

Language of instruction

Czech

Work placements

Not applicable.

Aims

The course aims to acquaint the students with the basics of algebraic operations, linear algebra, vector and Euclidean spaces, and analytic geometry. This will enable them to attend further mathematical and engineering courses and deal with engineering problems. Another goal of the course is to develop the students' logical
thinking.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended, attendance at seminars is required. The lessons are planned on the basis of a weekly schedule. The way of compensation for an absence is in the competence of the teacher.

Classification of course in study plans

  • Programme B-MAI-P Bachelor's, 1. year of study, winter semester, 6 credits, compulsory

  • Programme MITAI Master's

    specialization NADE , any year of study, winter semester, 6 credits, elective
    specialization NBIO , any year of study, winter semester, 6 credits, elective
    specialization NGRI , any year of study, winter semester, 6 credits, elective
    specialization NNET , any year of study, winter semester, 6 credits, elective
    specialization NVIZ , any year of study, winter semester, 6 credits, elective
    specialization NCPS , any year of study, winter semester, 6 credits, elective
    specialization NSEC , any year of study, winter semester, 6 credits, elective
    specialization NEMB , any year of study, winter semester, 6 credits, elective
    specialization NHPC , any year of study, winter semester, 6 credits, elective
    specialization NISD , any year of study, winter semester, 6 credits, elective
    specialization NIDE , any year of study, winter semester, 6 credits, elective
    specialization NISY , any year of study, winter semester, 6 credits, elective
    specialization NMAL , any year of study, winter semester, 6 credits, compulsory
    specialization NMAT , any year of study, winter semester, 6 credits, elective
    specialization NSEN , any year of study, winter semester, 6 credits, elective
    specialization NVER , any year of study, winter semester, 6 credits, elective
    specialization NSPE , any year of study, winter semester, 6 credits, compulsory

Type of course unit

 

Lecture

39 hours, optionally

Teacher / Lecturer

doc. RNDr. Miroslav Kureš, Ph.D.
doc. Mgr. Petr Vašík, Ph.D.

Syllabus

1. Relations, equivalences, orders, mappings, operations.
2. Number sets, fields.
3. Vector spaces, subspaces, homomorphisms. The linear dependence of vectors, the basis and dimension..
4. Matrices and determinants.
5. Systems of linear equations.
6. The charakteristic polynomial, eigen values, eugen vectors. Jordan normal form.
7. Dual vector spaces. Linear forms.
8. Bilinear and quadratic forms.
9. Schwarz inequality. Orthogonality. Gram-Schmidt process.
10. Inner, exterior, cross and triple products – relations and applications.
11. Affine and euclidean spaces. Geometry of linear objects.
12. Geometry of conics and quadrics.
13. Reserve

Exercise

22 hours, compulsory

Teacher / Lecturer

doc. RNDr. Miroslav Kureš, Ph.D.
Mgr. Jan Pavlík, Ph.D.

Syllabus

Week 1: Basics of mathematical logic and operations on sets.
Following weeks: Seminar related to the topic of the lecture given in the previous week.

Computer-assisted exercise

4 hours, compulsory

Teacher / Lecturer

doc. RNDr. Miroslav Kureš, Ph.D.

Syllabus

Seminars with computer support are organized according to current needs. They enables students to solve algorithmizable problems by computer algebra systems.