Course detail

# Linear Algebra

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.

Language of instruction

Czech

Number of ECTS credits

6

Mode of study

Not applicable.

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.

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

Course curriculum

Not applicable.

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic 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.

Classification of course in study plans

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

• Programme MITAI Master's

specialization NMAL , 0 year of study, winter semester, compulsory
specialization NSPE , 0 year of study, winter semester, compulsory
specialization NBIO , 0 year of study, winter semester, elective
specialization NSEN , 0 year of study, winter semester, elective
specialization NVIZ , 0 year of study, winter semester, elective
specialization NGRI , 0 year of study, winter semester, elective
specialization NISD , 0 year of study, winter semester, elective
specialization NSEC , 0 year of study, winter semester, elective
specialization NCPS , 0 year of study, winter semester, elective
specialization NHPC , 0 year of study, winter semester, elective
specialization NNET , 0 year of study, winter semester, elective
specialization NVER , 0 year of study, winter semester, elective
specialization NIDE , 0 year of study, winter semester, elective
specialization NEMB , 0 year of study, winter semester, elective
specialization NADE , 0 year of study, winter semester, elective
specialization NMAT , 0 year of study, winter semester, elective
specialization NISY , 0 year of study, winter semester, elective

#### Type of course unit

Lecture

39 hod., optionally

Teacher / Lecturer

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.
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 hod., compulsory

Teacher / Lecturer

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 hod., compulsory

Teacher / Lecturer

Syllabus

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