Course detail
Linear Algebra
FP-VlaPAcad. year: 2017/2018
The course deals with the 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
6
Mode of study
Not applicable.
Guarantor
Department
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
The course contains lectures that explain basic principles, problems and methodology of the discipline, and exercises that promote the practical knowledge of the subject presented in the lectures.
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
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. (EN)
Nicholson, W. K.: Elementary Linear Algebra with Applications, PWS... (EN)
Rektorys, K. a spol.: Přehled užité matematiky I., II., Prometheus... (CS)
Searle, S. R.: Matrix Algebra Useful for Statistics, Wiley 1982. (EN)
Thomas, G. B., Finney, R.L.: Calculus and Analytic Geometry, Addis... 2005 (EN)
Nicholson, W. K.: Elementary Linear Algebra with Applications, PWS... (EN)
Rektorys, K. a spol.: Přehled užité matematiky I., II., Prometheus... (CS)
Searle, S. R.: Matrix Algebra Useful for Statistics, Wiley 1982. (EN)
Thomas, G. B., Finney, R.L.: Calculus and Analytic Geometry, Addis... 2005 (EN)
Recommended reading
Horák, P.: Algebra a teoretická aritmetika, Masarykova univerzita (CS)
Horák, P., Janyška, J.: Analytická geometrie, Msarykova univerzita (CS)
Janyška, J., Sekaninová, A.: Analytická geometrie kuželoseček a kvadrik (CS)
Karásek, J., Skula, L.: Lineární algebra, Cerm 2005 (CS)
Mezník, I., Karásek, J., Miklíček, J.: Matematika I. pro strojní fakulty (CS)
Nedoma, J,: Matematika I. Cerm 2001 (CS)
Nedoma, J.: Matematika I., část první: Algebra a geometrie, PC-DIR (CS)
Procházka, L. a spol.: Algebra, Academia 1990 (CS)
Horák, P., Janyška, J.: Analytická geometrie, Msarykova univerzita (CS)
Janyška, J., Sekaninová, A.: Analytická geometrie kuželoseček a kvadrik (CS)
Karásek, J., Skula, L.: Lineární algebra, Cerm 2005 (CS)
Mezník, I., Karásek, J., Miklíček, J.: Matematika I. pro strojní fakulty (CS)
Nedoma, J,: Matematika I. Cerm 2001 (CS)
Nedoma, J.: Matematika I., část první: Algebra a geometrie, PC-DIR (CS)
Procházka, L. a spol.: Algebra, Academia 1990 (CS)
Classification of course in study plans
Type of course unit
Lecture
39 hod., optionally
Teacher / Lecturer
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.
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
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.
Following weeks: Seminar related to the topic of the lecture given in the previous week.