Course detail

# Vector and Matrix Algebra

FEKT-BPA-VMPAcad. year: 2022/2023

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

Learning outcomes of the course unit

- 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

Prerequisites

Co-requisites

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

Course curriculum

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. Matrices functions, matrix exponential, applications.

9. Introduction to numerical methods. Numerical methods for root finding (bisection method, Newton method, iterative method)

10. Numerical solution of systems of nonlinear equations. Systems 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 integration.

12. Numerical integration (trapezoidal and Simpson method).

13. Numerical solution of differential equations: initial problems (Euler method and its modifications, Runge-Kutta methods), boundary value problems (very briefly).

Work placements

Aims

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

Recommended optional programme components

Prerequisites and corequisites

Basic literature

NOVÁK M., Mathematics 3 (Numerical methods: Excercise book), 2014, FEEC BUT 2014 (EN)

HEFFERON, J., Linear Algebra (EN)

Recommended reading

eLearning

**eLearning:**currently opened course

#### Type of course unit

eLearning

**eLearning:**currently opened course