Course details

# Linear Algebra

ILG Acad. year 2020/2021 Winter semester 5 credits

Matrices and determinants. Systems of linear equations. Vector spaces and subspaces. Linear representation, coordinate transformation. Own values and own vectors. Quadratic forms and conics.

Guarantor

Deputy Guarantor

Language of instruction

Completion

Time span

Assessment points

Department

Lecturer

Instructor

Hliněná Dana, doc. RNDr., Ph.D. (DMAT FEEC BUT)

Lengál Ondřej, Ing., Ph.D. (DITS FIT BUT)

Svoboda Zdeněk, RNDr., CSc. (DMAT FEEC BUT)

Vážanová Gabriela, Mgr. (DMAT FEEC BUT)

Vítovec Jiří, Mgr., Ph.D. (DMAT FEEC BUT)

Subject specific learning outcomes and competences

The students will acquire an elementary knowledge of linear algebra and the ability to apply some of its basic methods in computer science.

Learning objectives

The students will get familiar with elementary knowledge of linear algebra, which is needed for informatics applications. Emphasis is placed on mastering the practical use of this knowledge to solve specific problems.

Why is the course taught

Linear algebra is one of the most important branches of mathematics for engineers, regardless of their specialization, as it deals with both specific computational procedures and abstract concepts, which are useful for describing technical problems. The knowledge gained in the course is applied by graduates where engineering problems are written in matrices, vectors and linear equations. The mastering of the basic concepts and their context will facilitate for further study and development of the chosen field.

Prerequisite kwnowledge and skills

Secondary school mathematics.

Study literature

- Bečvář, J., Lineární algebra, matfyzpress, Praha, 2005. (in Czech).
- Kovár, M., Maticový a tenzorový počet, FEKT VUT, Brno, 2013. (in Czech).
- Olšák, P., Úvod do algebry, zejména lineární. FEL ČVUT, Praha, 2007. (in Czech).

Fundamental literature

- Bečvář, J., Lineární algebra, matfyzpress, Praha, 2005. (in Czech).
- Bican, L., Lineární algebra, SNTL, Praha, 1979. (in Czech).
- Birkhoff, G., Mac Lane, S. Prehľad modernej algebry, Alfa, Bratislava, 1979. (in Slovak).
- Havel, V., Holenda, J.,
*Lineární algebra*, STNL, Praha 1984. (in Czech). - Hejný, M., Zaťko, V, Kršňák, P., Geometria, SPN, Bratislava, 1985. (in Slovak).
- Kolman B.,
*Elementary Linear Algebra*, Macmillan Publ. Comp., New York 1986. - Kolman B.,
*Introductory Linear Algebra*, Macmillan Publ. Comp., New York 1993. - Neri, F., Linear algebra for computational sciences and engineering, Springer, 2016.
- Olšák, P., Úvod do algebry, zejména lineární. FEL ČVUT, Praha, 2007. (in Czech).

Syllabus of lectures

- Systems of linear homogeneous and non-homogeneous equations. Gaussian elimination.
- Matrices and matrix operations. Rank of the matrix. Frobenius theorem.
- The
*determinant*of a square matrix. Inverse and adjoint matrices. The methods of computing the determinant.The Cramer's Rule. - Numerical solution of systems of linear equations, iterative methods.
- The vector space and its subspaces. The basis and the dimension. The coordinates of a vector relative to a given basis. The sum and intersection of vector spaces.
- The inner product. Orthonormal systems of vectors. Orthogonal projection and approximation. Gram-Schmidt orthogonalisation process.
- The transformation of the coordinates.
- Linear mappings of vector spaces. Matrices of linear transformations.
- Rotation, translation, symmetry and their matrices, homogeneous coordinates.
- The eigenvalues and eigenvectors. The orthogonal projections onto eigenspaces.
- Conic sections.
- Quadratic forms and their classification using sections.
- Quadratic forms and their classification using eigenvectors.

Syllabus of numerical exercises

Examples of tutorials are chosen to suitably complement the lectures.

Progress assessment

- Evaluation of the five written tests (max 40 points).

Controlled instruction

- Participation in lectures in this course is not controlled.
- The knowledge of students is tested at exercises; at five written tests for 8 points each, and at the final exam for 60 points.
- If a student can substantiate serious reasons for an absence from an exercise, (s)he can either attend the exercise with a different group (please inform the teacher about that) or ask his/her teacher for an alternative assignment to compensate for the lost points from the exercise.
- The passing boundary for ECTS assessment: 50 points.

Exam prerequisites

The minimal total score of 10 points gained out of the mid-term exams.

Schedule

Day | Type | Weeks | Room | Start | End | Lect.grp | Groups | Info |
---|---|---|---|---|---|---|---|---|

Mon | exam | 2020-12-21 | A112 A113 A218 D0206 D0207 D105 E104 E105 E112 G108 G202 | 10:00 | 11:50 | 1BIA 1BIB 2BIA 2BIB | Předtermín | |

Mon | exam | 2021-01-25 | A112 A113 D0206 D0207 D105 E104 E105 E112 G202 N104 N105 P108 T10/1.36 T12/2.173 T8/010 T8/020 T8/030 | 12:00 | 13:50 | 1BIA 1BIB 2BIA 2BIB | 2. oprava | |

Mon | exercise | lectures | T8/302 | 15:00 | 16:50 | 1BIA 1BIB 2BIA 2BIB | xx | Vážanová |

Mon | exercise | lectures | T8/312 | 15:00 | 16:50 | 1BIA 1BIB 2BIA 2BIB | xx | Hlavičková |

Mon | exercise | lectures | T8/302 | 17:00 | 18:50 | 1BIA 1BIB 2BIA 2BIB | xx | Vážanová |

Tue | exercise | lectures | A113 | 08:00 | 09:50 | 1BIA 1BIB 2BIA 2BIB | xx | Vážanová |

Tue | exercise | lectures | A113 | 10:00 | 11:50 | 1BIA 1BIB 2BIA 2BIB | xx | Vážanová |

Tue | lecture | lectures | T12/2.173 | 13:00 | 14:50 | 1BIB 2BIA 2BIB | xx | Hlavičková |

Wed | exam | 2021-01-06 | A112 A113 A218 C228 D0206 D0207 D105 G108 G202 L314 M103 M104 M105 N103 N104 N105 N203 N204 N205 O204 | 08:00 | 09:50 | 1BIA 1BIB 2BIA 2BIB | řádná | |

Wed | exam | 2021-01-06 | A112 A113 A218 C228 D0206 D0207 D105 E104 E105 E112 G108 G202 L314 M103 M104 M105 N103 N104 N105 N203 N204 N205 O204 T10/1.36 T12/2.173 T12/SF1.141 T8/010 T8/020 T8/030 | 12:00 | 13:50 | 1BIA 1BIB 2BIA 2BIB | řádná | |

Wed | lecture | lectures | D0207 D105 | 12:00 | 13:50 | 1BIA 2BIA 2BIB | xx | Hliněná |

Wed | lecture | 2., 3., 4., 5., 6., 7., 8., 9., 10., 11., 12., 13. of lectures | D105v | 12:00 | 13:50 | YT, ZP; Hliněná | ||

Wed | exercise | lectures | D0207 | 14:00 | 15:50 | 1BIA 1BIB 2BIA 2BIB | xx | Hliněná |

Wed | exercise | lectures | A113 | 16:00 | 17:50 | 1BIA 1BIB 2BIA 2BIB | xx | Hlavičková |

Wed | exercise | 2., 3., 4., 6. of lectures | A113v | 16:00 | 17:50 | YT, ZP; Hlavičková | ||

Wed | exercise | lectures | D0207 | 16:00 | 17:50 | 1BIA 1BIB 2BIA 2BIB | xx | Hliněná |

Wed | exam | 2021-01-06 | A112 A113 A218 C228 D0206 D0207 D105 E104 E105 E112 G108 G202 L314 M103 M104 M105 N103 N104 N105 N203 N204 N205 O204 | 18:00 | 19:50 | 1BIA 1BIB 2BIA 2BIB | řádná | |

Thu | exercise | lectures | A113 | 10:00 | 11:50 | 1BIA 1BIB 2BIA 2BIB | xx | Hliněná |

Thu | exercise | 2., 3., 4., 5., 7., 8., 9., 10., 11., 12., 13. of lectures | A113v | 10:00 | 11:50 | YT, ZP, bez projekce; Hliněná | ||

Thu | exercise | lectures | D0207 | 12:00 | 13:50 | 1BIA 1BIB 2BIA 2BIB | xx | Vážanová |

Thu | exercise | lectures | A113 | 14:00 | 15:50 | 1BIA 1BIB 2BIA 2BIB | xx | Hliněná |

Thu | exercise | 2., 3., 4., 5., 7., 8., 9., 10., 11., 12., 13. of lectures | A113v | 14:00 | 15:50 | YT, ZP, bez projekce; Hliněná | ||

Thu | exercise | lectures | D0207 | 14:00 | 15:50 | 1BIA 1BIB 2BIA 2BIB | xx | Vážanová |

Fri | exercise | lectures | T8/020 | 07:00 | 08:50 | 1BIA 1BIB 2BIA 2BIB | xx | Vítovec |

Fri | exercise | lectures | T8/020 | 09:00 | 10:50 | 1BIA 1BIB 2BIA 2BIB | xx | Vítovec |

Fri | exam | 2021-01-15 | A112 A113 A218 C228 D0206 D0207 D105 E104 E105 E112 G108 G202 L314 M103 M104 M105 N103 N104 N105 N203 N204 N205 O204 T10/1.36 T12/2.162 T12/2.173 T12/SF1.141 T8/010 T8/020 T8/030 | 12:00 | 13:50 | 1BIA 1BIB 2BIA 2BIB | 1. oprava |

Course inclusion in study plans