Course detail

# Linear Algebra

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

Learning outcomes of the course unit

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

Prerequisites

Secondary school mathematics.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Kovár, M.,  Maticový a tenzorový počet, FEKT VUT, Brno, 2013. (CS)
Bečvář, J., Lineární algebra, matfyzpress, Praha, 2005 (CS)
Bican, L., Lineární algebra, SNTL, Praha, 1979 (CS)
Birkhoff, G., Mac Lane, S. Prehľad modernej algebry, Alfa, Bratislava, 1979 (CS)
Havel, V., Holenda, J., Lineární algebra, STNL, Praha 1984. (CS)
Hejný, M., Zaťko, V, Kršňák, P., Geometria, SPN, Bratislava, 1985 (CS)
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.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

• Evaluation of two homework assignments - groupwork (max 10 points).
• Evaluation of the two mid-term exams (max 30 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.

Language of instruction

Czech, English

Work placements

Not applicable.

Aims

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.

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

• 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.

Classification of course in study plans

• Programme BIT Bachelor's, 1. year of study, winter semester, 5 credits, compulsory

• Programme IT-BC-3 Bachelor's

branch BIT , 1. year of study, winter semester, 5 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

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.

Computer-assisted exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

Examples of tutorials are chosen to suitably complement the lectures.