Course detail

# Linear Algebra

FSI-SLAAcad. year: 2018/2019

The course deals with following topics: Sets: mappings of sets, relations on a set.

Algebraic operations: groupoids, 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.

Supervisor

Department

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.

Recommended or required reading

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.

Nicholson, W. K.: Elementary Linear Algebra with Applications, PWS 1990.

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.

Form of examinations: The examination has a written and an oral part. In a 120-minute written test, students

solve the following 5 problems.

During the oral part of the examination, the examiner goes through the test with the student. The examiner should inform the students at the last lecture about the basic rules of the examination and the evaluation of its results.

Rules for classification: Student can achieve 4 points for each problem. Therefore, the students may achieve 20 points in total.

Final classification: A (excellent): 19 to 20 points

B (very good): 17 to 18 points

C (good): 15 to 16 points

D (satisfactory): 13 to 14 points

E (sufficient): 10 to 12 points

F (failed): 0 to 9 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.

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

1. week. Relations, equivalences, orders, mappings, operations, algebraic structures, fields.

2. week. Vector spaces, subspaces, homomorphisms. The linear dependence of vectors, the basis and dimension..

3. week. Matrices and determinants.

4. week. Systems of linear equations.

5. week. The charakteristic polynomial, eigen values, eugen vectors. Jordan normal form.

6. week. Dual vector spaces. Linear forms.

7. week. Bilinear and quadratic forms.

8. week. Unitary spaces. Schwarz inequality. Orthogonality.

9. week. Inner, exterior, cross and triple products – relations and applications.

10. week. Symplectic spaces.

11. week. Affine and euclidean spaces. Geometry of linear objects.

12. week. Projective spaces.

13. week. Geometry of conics and quadrics.

Exercise

22 hours, 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 hours, 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.

eLearning

**eLearning:** currently opened course