Course details

# Linear Algebra

SLA FME BUT SLA Acad. year 2019/2020 Winter semester 6 credits

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.

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Subject specific 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.

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.

Prerequisite kwnowledge and skills

Students are expected to have basic knowledge of secondary school mathematics.

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)

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.

Progress assessment

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 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.

Exam prerequisites

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.

Schedule

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