Course detail

# Vector and Matrix Algebra

FEKT-BPC-VMPAcad. year: 2024/2025

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

Number of ECTS credits

Mode of study

Guarantor

Department

Entry knowledge

Rules for evaluation and completion of the course

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

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

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

Elearning

**eLearning:**currently opened course

Classification of course in study plans

#### Type of course unit

Lecture

Teacher / Lecturer

Syllabus

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

Teacher / Lecturer

Syllabus

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

Teacher / Lecturer

Syllabus

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