Course detail

Vector and Matrix Algebra

FEKT-BPC-VMPAcad. year: 2023/2024

In the part of vector calculus, attention is focused on vector spaces, basic concepts, linear combinations of vectors, linear dependence, independence of vectors, bases, dimensions of vector space. The introduction of a scalar product makes it possible to explain the orthogonalization of vectors and to search for the orthogonal projection of a vector on a subspace and to apply this knowledge to the solution of predetermined systems and the least squares method. In the part of matrix calculus, students are introduced to matrix algebra, eigenvalues and eigenvectors are studied and their use to diagnose matrices and calculate matrix functions and their applications. Furthermore, the positive-definite of matrices is studied.
The part of numerical methods discusses the solution of nonlinear equations and matrix systems of linear equations, approximation of functions using an interpolation polynomial, spline and least squares method, 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 the knowledge of vector calculus in a real field and calculate in a complex field at the secondary school level. Similarly, he should be able to use the basics of mathematical analysis at the secondary school leve

Rules for evaluation and completion of the course

The student's work during the semestr (written tests and homework) is assessed by maximum 30 points. Written examination is evaluated by maximum 70 points. It consist of several tasks (half of them in matrix calculation and the second half in numerical methods) and two theoretical questions (1+1, each for 5 points). To pass the exam, the student must gain at least 10 points in matrix calculation and at least 10 points in numerical methods.
The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.

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

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)
M. Dont: Maticová analýza, skripta, nakl. ČVUT 2011 (CS)

Recommended reading

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

eLearning

eLearning: currently opened course

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 hours, 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 hours, 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 hours, 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.

eLearning

eLearning: currently opened course