Fundamentals of Linear Algebra (FSI-TLA)

Academic year 2018/2019
Supervisor: doc. Mgr. Jaroslav Hrdina, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
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.
Learning outcomes and competences:
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.
Course contents:
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.
Teaching methods and criteria:
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
Controlled participation in lessons:
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  13 × 2 hrs. optionally                  
    Exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture 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 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.
Literature - fundamental:
1. 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
2. KARÁSEK, J., SKULA, L.: Lineární Algebra. Brno: AKADEMICKÉ NAKLADA-. TELSTVÍ CERM, 2005. 179 p. ISBN 80-214-3100-8.
3. Lang, Serge (March 9, 2004), Linear Algebra, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-96412-6
4. AXLER, S. J. (1997). Linear algebra done right. New York, Springer.
Literature - recommended:
5. Horák, P., Janyška, J.: Analytická geometrie, Masarykova univerzita 1997
6. Janyška, J., Sekaninová, A.: Analytická teorie kuželoseček a kvadrik, Masarykova univerzita 1996
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B3A-P full-time study B-FIN Physical Engineering and Nanotechnology -- Cr,Ex 3 Compulsory 1 1 W