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

Czech

Completion

Credit+Examination (written)

Time span

26 hrs lectures, 26 hrs exercises

Assessment points

60 exam, 30 half-term test, 10 projects

Department

Department of Mathematics (DMAT FEEC BUT)

Lecturer

Instructor

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

1. Systems of linear homogeneous and non-homogeneous equations. Gaussian elimination.
2. Matrices and matrix operations. Rank of the matrix. Frobenius theorem.
3. The determinant of a square matrix. Inverse and adjoint matrices. The methods of computing the determinant.The Cramer's Rule.
4. 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.
5. The inner product. Orthonormal systems of vectors. Orthogonal projection and approximation. Gram-Schmidt orthogonalisation process.
6. The transformation of the coordinates.
7. Linear mappings of vector spaces. Matrices of linear transformations.
8. Rotation, translation, symmetry and their matrices, homogeneous coordinates.
9. The eigenvalues and eigenvectors. The orthogonal projections onto eigenspaces.
10. Numerical solution of systems of linear equations, iterative methods.
11. Conic sections.
12. Quadratic forms and their classification using sections.
13. 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 two homework assignments - groupwork (max 10 points).
• Evaluation of the two mid-term exams (max 30 points).

Controlled instruction

• Participation in lectures in this course is not controlled.
• The knowledge of students is tested at exercises; including two homework assignments worth for 5 points each, at two midterm exams for 15 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. Plagiarism and not allowed cooperation will cause that involved students are not classified and disciplinary action may be initiated.

Course inclusion in study plans

• Programme BIT, 1st year of study, Compulsory
• Programme IT-BC-3, field BIT, 1st year of study, Compulsory