Course detail

# Fundamentals of Linear Algebra

FSI-TLAAcad. year: 2017/2018

The course deals with the following topics:

Algebraic operations: groupoids, semigroups, groups, vector spaces, matrices and operations on matrices.

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 of a square matrix, diagonalization.

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

Jan Slovák, Martin Panák, Michal Bulant a kolektiv
Matematika drsně a svižně, 1. vyd. — Brno : Masarykova univerzita, 2013 — 773 s. , Jan Slovák, Martin Panák, Michal Bulant a kolektiv
ISBN 978-80-210-6307-5 (CS)

KARÁSEK, J., SKULA, L.: Lineární Algebra. Brno: AKADEMICKÉ NAKLADA-. TELSTVÍ CERM, 2005. 179 p. ISBN 80-214-3100-8. (CS)

Lang, Serge (March 9, 2004), Linear Algebra, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-96412-6
(EN)

AXLER, S. J. (1997). Linear algebra done right. New York, Springer.
(EN)

Horák, P., Janyška, J.: Analytická geometrie, Masarykova univerzita 1997 (CS)

Janyška, J., Sekaninová, A.: Analytická teorie kuželoseček a kvadrik, Masarykova univerzita 1996 (CS)

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:

Problem 1: Groupoids, vector spaces, euclidean spaces, eigenvalues and eigenvectors.

Problem 2: Matrices.

Problem 3: Systems of linear equations.

Problem 4: Analytic geometry of linear concepts.

Problem 5: Analytic geometry of nonlinear concepts.

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: The student can achieve 4 points for each problem. Therefore he/she 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 staces, and analytic geometry. This will enable them to attend further mathematical and engineering courses and deal with engineering problems. Another goal of the copurse 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

26 hours, optionally

Teacher / Lecturer

Syllabus

1. Algebraic operations: groupoids, subgroupoids, semigroups, neutral element, inverse element.

2. Groups, subgroups.

3. Vector spaces: definition, linear combination, linear independence.

4. Vector subspace, basis and dimension of a vector space.

5. Matrices and operations on matrices. Rings, commutative rings, zero divisors.

6. Linear algebra: determinants, Cauchy´s theorem, inverse matrix.

7. Matrices in step form, rank of a matrix.

8. Systems of linear equations: Cramer´s rule, elimination method, Frobenius´s theorem, homogeneous systems.

9. Euclidean spaces: scalar product, norm, Schwarz inequality, Gram-Schmidt orthogonalization algorithm.

10. Eigenvalues and eigenvectors of a square matrix, characteristic polynomial, diagonalization. Fundamentals of analytic geometry: cross and mixed products of vectors.

11. Analytic geometry of linear concepts.

12. Analytic geometry of conics.

13. Analytic geometry of quadrics.

Exercise

13 hours, compulsory

Teacher / Lecturer

Syllabus

1st week: Basics of set theory, operations on sets, mappings.

Following weeks: Seminar related to the topic of the lecture given in the previous week.