Course detail

Linear Algebra

FSI-SLAAcad. year: 2011/2012

The course deals with following topics: Sets: mappings of sets, relations on a set.
Algebraic operations: groupoids, 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.

Language of instruction

Czech

Number of ECTS credits

7

Mode of study

Not applicable.

Guarantor

doc. RNDr. Jiří Karásek, CSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

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.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

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: Mappings, 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: 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

Course curriculum

Not applicable.

Work placements

Not applicable.

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.

Specification of controlled education, way of implementation and compensation for absences

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Howard, A. A.: Elementary Linear Algebra, Wiley 2002.
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.

Classification of course in study plans

  • Programme B3901-3 Bachelor's

    branch B-MAI , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

doc. RNDr. Jiří Karásek, CSc.

Syllabus

Week 1: Fundamentals of mappings of sets: concept of a mapping, injection, surjection and bijection, inverse
mapping, composition of mappings.
Concept of a relation: general definition, reflexive, symmetric, antisymmetric, transitive and complete
relation, order, linear order.
Week 2: Equivalence, decomposition of a set, relationship between an equivalence and a decomposition.
Algebraic operations: groupoid, subgroupoid, semigroup, neutral element, inverse element.
Week 3: Group, subgroup.
Vector spaces: definition, linear combination, linear independence.
Week 4: Vector subspace, basis and dimension of a vector space.
Week 5: Matrices and operations on matrices.
Rings, Commutative rings, zero divisors.
Week 6: Fundamentals of linear algebra: determinants, Cauchy´s theorem, inverse matrix.
Week 7: Matrices in step form, rank of a matrix.
Week 8: Systems of linear equations: Cramer´s rule, elimination method, Frobenius´s theorem, homogeneous
systems.
Week 9: Euclidean spaces: scalar product, norm, Schwarz inequality, Gram-Schmidt orthogonalization algorithm.
Week 10: Eigenvalues and eigenvectors, characteristic polynomial.
Fundamentals of analytic geometry: cross and mixed product of vectors.
Week 11: Analytic geometry of linear concepts.
Week 12: Analytic geometry of conics.
Week 13: Analytic geometry of quadrics.

Exercise

26 hod., compulsory

Teacher / Lecturer

doc. RNDr. Jiří Karásek, CSc.

Syllabus

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.