Linear Algebra

• 26 hrs lectures
• 26 hrs exercises

• 60 pts final exam (written part)
• 40 pts mid-term test (written part)

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

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.

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.

Secondary school mathematics.

• Kovár, M.,  Maticový a tenzorový počet, FEKT VUT, Brno, 2013. (in Czech).
• 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).

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. Numerical solution of systems of linear equations, iterative methods.
5. 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.
6. The inner product. Orthonormal systems of vectors. Orthogonal projection and approximation. Gram-Schmidt orthogonalisation process.
7. The transformation of the coordinates.
8. Linear mappings of vector spaces. Matrices of linear transformations.
9. Rotation, translation, symmetry and their matrices, homogeneous coordinates. 
10. The eigenvalues and eigenvectors. The orthogonal projections onto eigenspaces.
11. Conic sections.
12. Quadratic forms and their classification using sections.
13. Quadratic forms and their classification using eigenvectors.

Examples of tutorials are chosen to suitably complement the lectures.

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

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

• 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.

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