Linear Algebra (FSI-SLA)

Academic year 2022/2023
Supervisor: doc. RNDr. Miroslav Kureš, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Course type: departmental course, applied basis
Aims of the course unit:
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.
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 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.
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, 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
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 × 3 hrs. optionally                  
    Exercise  11 × 2 hrs. compulsory                  
    Computer-assisted exercise  2 × 2 hrs. compulsory                  
Course curriculum:
    Lecture 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 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 Seminars with computer support are organized according to current needs. They enables students to solve algorithmizable problems by computer algebra systems.
Literature - fundamental:
2. Thomas, G. B., Finney, R.L.: Calculus and Analytic Geometry, Addison Wesley 2003.
3. Howard, A. A.: Elementary Linear Algebra, Wiley 2002.
4. Rektorys, K. a spol.: Přehled užité matematiky I., II., Prometheus 1995.
5. Searle, S. R.: Matrix Algebra Useful for Statistics, Wiley 1982.
Literature - recommended:
6. Karásek, J., Skula, L.: Algebra a geometrie, Cerm 2002.
7. Nedoma, J.: Matematika I., Cerm 2001.
8. Nedoma, J.: Matematika I., část první: Algebra a geometrie, PC-DIR 1998.
9. Horák, P., Janyška, J.: Analytická geometrie, Masarykova univerzita 1997.
10. Janyška, J., Sekaninová, A.: Analytická teorie kuželoseček a kvadrik, Masarykova univerzita 1996.
11. Mezník, I., Karásek, J., Miklíček, J.: Matematika I. pro strojní fakulty, SNTL 1992.
12. Horák, P.: Algebra a teoretická aritmetika, Masarykova univerzita 1991.
13. Procházka, L. a spol.: Algebra, Academia 1990.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
CŽV full-time study CZV Bases of Mechanical Engineering -- Cr,Ex 6 Compulsory 1 1 W
MITAI full-time study NEMB do 2021/22 Embedded Systems -- Cr,Ex 6 Elective 1 0 W
B-MAI-P full-time study --- no specialisation -- Cr,Ex 6 Compulsory 1 1 W
MITAI full-time study NMAL Machine Learning -- Cr,Ex 6 Compulsory 1 0 W
MITAI full-time study NSPE Sound, Speech and Natural Language Processing -- Cr,Ex 6 Compulsory 1 0 W
MITAI full-time study NADE Application Development -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NBIO Bioinformatics and Biocomputing -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NCPS Cyberphysical Systems -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NEMB Embedded Systems -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NGRI Computer Graphics and Interaction -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NHPC High Performance Computing -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NIDE Intelligent Devices -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NISD Information Systems and Databases -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NISY Intelligent Systems -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NISY do 2020/21 Intelligent Systems -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NMAT Mathematical Methods -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NNET Computer Networks -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NSEC Cybersecurity -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NSEN Software Engineering -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NVER Software Verification and Testing -- Cr,Ex 6 Elective 1 0 W
MITAI full-time study NVIZ Computer Vision -- Cr,Ex 6 Elective 1 0 W