Course detail
Vector and Matrix Algebra
FEKT-BPA-VMPAcad. year: 2024/2025
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
Number of ECTS credits
Mode of study
Guarantor
Department
Offered to foreign students
Entry knowledge
Rules for evaluation and completion of the course
Aims
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
Prerequisites and corequisites
Basic literature
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
Elearning
Classification of course in study plans
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
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
Elearning