Course detail

Vector and Matrix Algebra

FEKT-BPA-VMPAcad. year: 2025/2026

In the field of matrix claculus, main attention is paid to vector spaces, basic notions, linear combination of vectors, linear dependence, independence vectors, base, dimension of a vector space, matrix algebra, eigenvalues and eigenvectors, matrix functions and their applications.
In the field of numerical mathematics, the following topics are covered: root finding,matrices systems of linear equations, convergence analysis, curve fitting (interpolation and splines, least squares method), numerical differentiation and integration.

Language of instruction

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Michal Novák, Ph.D.

Department

Department of Mathematics (UMAT)

Offered to foreign students

Of all faculties

Entry knowledge

The student should be able to apply the basic knowledge of verctor calculus in real and complex domain on the secondary school level

Rules for evaluation and completion of the course

Work during semester is awarded by maximum 30 points for 1 test written at the end of the semester. Final exam is awarded by maximum 70 points.

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 ( v reálném i komplexním oboru)
- 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 type of a matrix 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
- find the approximate solution of a differential equation using Euler method, modified Euler methods and Runge-Kutta methods

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

HEFFERON, J., Linear Algebra (EN)
M. Novák, Linear Algebra and Related Issues, Brno, VUT, 2022 (EN)
NOVÁK M., Mathematics 3 (Numerical methods: Excercise book), 2014, FEEC BUT 2014 (EN)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme BPA-ELE Bachelor's

    specialization BPA-ECT , 1 year of study, winter semester, compulsory
    specialization BPA-PSA , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Syllabus

1. Vectors, vector spaces.
2. Matrices, algebra of matrices, determinant.
3. Linear Systems.
4.Eigenvalues, eigenvectors.
5. Orthogonal matrices.
6. Hermitian and unitary matrices.
7. Definitness of matrices.
8. Introduction to numerical methods, numerical solution of non-linear equations.
9. Numerical solution of nonlinear systems and linear systems.
10. Interpolaton and interpolation polynomial, splines.
11. Least squares methos, numerical differentiation.
12. Numerical integration. 
13. Revision

Fundamentals seminar

12 hod., compulsory

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Exercise in computer lab

14 hod., compulsory

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.