Title:

# Linear Algebra

Code:SLA (FSI SLA)
Ac.Year:2019/2020
Sem:Winter
Curriculums:
ProgrammeField/
Specialization
YearDuty
MITAINBIO-Elective
MITAINCPS-Elective
MITAINEMB-Elective
MITAINGRI-Elective
MITAINHPC-Elective
MITAINIDE-Elective
MITAINISD-Elective
MITAINISY-Elective
MITAINMAL-Compulsory
MITAINMAT-Elective
MITAINNET-Elective
MITAINSEC-Elective
MITAINSEN-Elective
MITAINSPE-Compulsory
MITAINVER-Elective
MITAINVIZ-Elective
Language of Instruction:Czech
Credits:6
Completion:credit+exam (written)
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:3922040
ExamsTestsExercisesLaboratoriesOther
Points:60202000
Guarantor:Kureš Miroslav, Doc. RNDr., Ph.D. (DADM)
Deputy guarantor:Černocký Jan, doc. Dr. Ing. (DCGM)
Lecturer:Kureš Miroslav, Doc. RNDr., Ph.D. (DADM)
Instructor:Kureš Miroslav, Doc. RNDr., Ph.D. (DADM)
Faculty:Faculty of Mechanical Engineering BUT
Department:Department of Algebra and Discrete Mathematics FME BUT

Learning objectives:
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.
Description:
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.
Knowledge and skills required for the course:
Students are expected to have basic knowledge of secondary school mathematics.
Learning outcomes and competencies:
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.
Syllabus of lectures:
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 characteristic polynomial, Eugen 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.
Syllabus of numerical exercises:
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.
Syllabus of computer exercises:
Seminars with computer support are organized according to current needs. They enable students to solve algorithmizable problems with computer algebra systems.
Study 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. (in Czech)
• 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. (in Czech)
• Nedoma, J.: Matematika I., Cerm 2001. (in Czech)
• Nedoma, J.: Matematika I., část první: Algebra a geometrie, PC-DIR 1998. (in Czech)
• Horák, P., Janyška, J.: Analytická geometrie, Masarykova univerzita 1997. (in Czech)
• Janyška, J., Sekaninová, A.: Analytická teorie kuželoseček a kvadrik, Masarykova univerzita 1996.(in Czech)
• Mezník, I., Karásek, J., Miklíček, J.: Matematika I. pro strojní fakulty, SNTL 1992. (in Czech)
• Horák, P.: Algebra a teoretická aritmetika, Masarykova univerzita 1991. (in Czech)
• Procházka, L. a spol.: Algebra, Academia 1990. (in Czech)
Controlled instruction:
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.

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Progress assessment:
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