Course detail

Vector and Matrix Algebra

FEKT-BPC-VMPAcad. year: 2025/2026

In the vector calculus section, the focus is on vector spaces, basic concepts, linear combinations of vectors, linear dependence, independence of vectors, bases, dimensions of vector space. The introduction of the scalar product allows to explain orthogonalization of vectors and to find the orthogonal projection of a vector onto a subspace and to apply this knowledge to the solution of overdetermined systems and the least squares method. In the matrix calculus section, students are introduced to matrix algebra, eigenvalues and eigenvectors are studied and their use to diagonalize matrices and calculate matrix functions and their application, focusing on the exponential of a matrix . The definiteness of matrices is also discussed. The numerical methods section discusses the solution of nonlinear equations and matrix systems of linear equations, approximation of functions by interpolating polynomial, spline and least squares methods, numerical derivation and integration. 

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Zdeněk Svoboda, CSc.

Department

Department of Mathematics (UMAT)

Entry knowledge

The student should be able to apply knowledge of vector calculus in a real-world domain and calculate in a complex domain at the high school level. Similarly, the student should be able to apply basic mathematical analysis at the high school level 

Rules for evaluation and completion of the course

The work during the semester is assessed with a maximum of 30 points (these points can be obtained for tests, homework). In addition, a maximum of 10 bonus points can be earned for bonus homework, additional theoretical questions and increased activity in practical exercises. Students must earn at least 10 points to receive credit. The final written examination is marked with a maximum of 70 points. It consists of examples verifying the mastery of algorithms and theoretical questions (total of 5 points in matrix calculus and 5 points in numerical methods). To pass the exam, the student must obtain at least 10 points in the matrix calculations section and at least 10 points in the numerical methods section. The definition of supervised learning and the way it is carried out are given in the annually updated regulation of the course guarantor.

Aims

The aim of this course is to introduce the basics of vector and matrix calculus in real and complex domain and basic numerical solution methods of systems of equations.
Students completing this course should be able to:
- decide whether vectors are linearly independent and whether they form a basis of a vector space (in the real and complex field)
- add and multiply matrices, compute the determinant of a square matrix to the 4x4 type, compute the rank and the inverse of a matrix
- solve a system of linear equations
- compute eigenvalues and eigenvectors of a matrix
- analyze the definiteness of a matrix also using eigenvalues
- compute a matrix exponential for certain classes of matrices
- solve matrices systems of linear equations
- find the root of a given equation f(x)=0 using the bisection method, Newton method or the iterative method, describe these methods including the convergence conditions
- solve a system of linear equations using Gaussian elimination with pivoting, Jacobi and Gauss-Seidel iteration methods, discuss the advantages and disadvantages of these methods
- find the approximation of a function by spline functions
- find the approximation of a function given by table of points by the least squares method (linear, quadratic or exponential approximation)
- estimate the derivative of a given function using numerical differentiation
- compute the numerical approximation of a definite integral using trapezoidal and Simpson method, describe the principal of these methods, compare them according to their accuracy

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

M. Dont: Maticová analýza, skripta, nakl. ČVUT 2011 (CS)
FAJMON, B., HLAVIČKOVÁ, I., NOVÁK, M., Matematika 3. Elektronický text FEKT VUT, Brno, 2013 (CS)
SCHMIDTMAYER, J., Maticový počet a jeho použití v technice, SNTL Praha 1974 (CS)

Recommended reading

HRUZA, B., MRHAČOVÁ, H., Cvičení z algebry a geometrie. Ediční stř. VUT 1993, skriptum. (CS)

Classification of course in study plans

  • Programme BPC-AMT Bachelor's 1 year of study, winter semester, compulsory
  • Programme BPC-SEE Bachelor's 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Zdeněk Svoboda, CSc.

Syllabus

1. Vectors, vector spaces.
2. Matrices, matrix algebra, determinant of a matrix
3. Systems of linear equations.
4. Eigenvalues and eigenvectors of a matrix.
5. Ortogonalization, ortogonal projection.
6. Hermitian a unitary matrix.
7. Definite matrices, characteristic using eigenvalues.
8. Matrix functions, matrix exponential, applications.
9. Introduction to numerical methods. Numerical methods for root finding (bisection method, Newton method, iterative method)
10. Numerical solution of linear equations (Gaussian elimination with pivoting, Jacobi and Gauss-Seidel iterative methods).
11. Interpolation: interpolation polynomial (Lagrange and Newton), splines (linear and cubic)
12. Least squares method. Numerical differentiation.
13. Numerical differentiation and integration.

Fundamentals seminar

12 hod., compulsory

Teacher / Lecturer

doc. RNDr. Zdeněk Svoboda, CSc.

Syllabus

1. Vectors, Matrices, matrix algebra, determinant of a matrix
2. Systems of linear equations, eigenvalues and eigenvectors of a matrix.
3. Ortogonalization, ortogonal projection.
4. Definite matrices, characteristic using eigenvalues.matrix functions, matrix exponential, applications.
5. Numerical methods for root finding linear and nonlinear equations
6. Interpolation: interpolation polynomial east squares method. Numerical differentiation and integration.

Computer-assisted exercise

14 hod., compulsory

Teacher / Lecturer

doc. RNDr. Zdeněk Svoboda, CSc.

Syllabus

1. Vectors, Matrices, matrix algebra, determinant of a matrix
2. Systems of linear equations, eigenvalues and eigenvectors of a matrix.
3. Ortogonalization, ortogonal projection.
4. Definite matrices, characteristic using eigenvalues.matrix functions, matrix exponential, applications.
5. Numerical methods for root finding linear and nonlinear equations
6. Interpolation: interpolation polynomial east squares method.
7.Numerical differentiation and integration.