Course detail

Linear Algebra I

FSI-SLAAcad. year: 2023/2024

The course deals with the following topics: Vector spaces, matrices and operations on matrices. Consequently, determinants, matrices in a step form and the rank of a matrix, systems of linear equations. Euclidean spaces: scalar product of vectors, eigenvalues and eigenvectors, Jordan canonical form. Fundamentals of analytic geometry, linear objects.

Language of instruction

Czech

Number of ECTS credits

6

Mode of study

Not applicable.

Guarantor

doc. Mgr. Petr Vašík, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

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

Rules for evaluation and completion of the course

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

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.

Aims

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.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Howard, A. A.: Elementary Linear Algebra, Wiley 2002.
Slovák J., Lineární algebra, Masarykova univerzita, http://www.math.muni.cz/~slovak/ftp/lectures/linearni.algebra/ (CS)
Rektorys, K. a spol.: Přehled užité matematiky I., II., Prometheus 1995.
Searle, S. R.: Matrix Algebra Useful for Statistics, Wiley 1982.
Thomas, G. B., Finney, R.L.: Calculus and Analytic Geometry, Addison Wesley 2003.

Recommended reading

Horák, P.: Algebra a teoretická aritmetika, Masarykova univerzita 1991.
Horák, P., Janyška, J.: Analytická geometrie, Masarykova univerzita 1997.
Janyška, J., Sekaninová, A.: Analytická teorie kuželoseček a kvadrik, Masarykova univerzita 1996.
Karásek, J., Skula, L.: Algebra a geometrie, Cerm 2002.
Mezník, I., Karásek, J., Miklíček, J.: Matematika I. pro strojní fakulty, SNTL 1992.
Nedoma, J.: Matematika I., Cerm 2001.
Nedoma, J.: Matematika I., část první: Algebra a geometrie, PC-DIR 1998.
Procházka, L. a spol.: Algebra, Academia 1990.

Elearning

eLearning: currently opened course

Classification of course in study plans

  • Programme B-MAI-P Bachelor's 1 year of study, winter semester, compulsory

  • Programme MITAI Master's

    specialization NSPE , 0 year of study, winter semester, compulsory
    specialization NBIO , 0 year of study, winter semester, elective
    specialization NSEN , 0 year of study, winter semester, elective
    specialization NVIZ , 0 year of study, winter semester, elective
    specialization NGRI , 0 year of study, winter semester, elective
    specialization NADE , 0 year of study, winter semester, elective
    specialization NISD , 0 year of study, winter semester, elective
    specialization NMAT , 0 year of study, winter semester, elective
    specialization NSEC , 0 year of study, winter semester, elective
    specialization NISY up to 2020/21 , 0 year of study, winter semester, elective
    specialization NCPS , 0 year of study, winter semester, elective
    specialization NHPC , 0 year of study, winter semester, elective
    specialization NNET , 0 year of study, winter semester, elective
    specialization NMAL , 0 year of study, winter semester, compulsory
    specialization NVER , 0 year of study, winter semester, elective
    specialization NIDE , 0 year of study, winter semester, elective
    specialization NEMB , 0 year of study, winter semester, elective
    specialization NISY , 0 year of study, winter semester, elective
    specialization NEMB up to 2021/22 , 0 year of study, winter semester, elective

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

doc. Mgr. Petr Vašík, Ph.D.

Syllabus

1. Number sets, fields. Vector spaces, subspaces, homomorphisms.
2.  Linear dependence of vectors, basis and dimension.
3. Matrices and determinants.
4. Systems of linear equations.
5. The characteristic polynomial, eigen values, eigen vectors. 
6. Jordan normal form.
7. Euclidean and unitary vector spaces.
8. Dual vector spaces. Linear forms.
9. Bilinear and quadratic forms.
10. Orthogonality. Gram-Schmidt process.
11. Inner, exterior and cross products – relations and applications.
12. Affine and euclidean spaces. Geometry of linear objects.
13. Reserve

Exercise

22 hod., compulsory

Teacher / Lecturer

doc. Mgr. et Mgr. Aleš Návrat, Ph.D.
doc. Mgr. Petr Vašík, Ph.D.

Syllabus

Week 1: Fundamental notions such as vectors, matrices and operations.
Following weeks: Seminar related to the topic of the lecture given in the previous week.

Computer-assisted exercise

4 hod., compulsory

Teacher / Lecturer

doc. Mgr. Petr Vašík, Ph.D.

Syllabus

Seminars with computer support are organized according to current needs. They enables students to solve algorithmizable problems by computer algebra systems.

Elearning

eLearning: currently opened course