Vector and Matrix Algebra
FEKT-BPC-VMPAcad. year: 2018/2019
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.
Learning outcomes of the course unit
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
The student should be able to apply the basic knowledge of verctor calculus in real and complex domain on the secondary school level
Recommended optional programme components
Recommended or required reading
FAJMON, B., HLAVIČKOVÁ, I., NOVÁK, M., Matematika 3. Elektronický text FEKT VUT, Brno, 2013 (CS)
Planned learning activities and teaching methods
Teaching methods include lectures, computer and numerical exercises.
Assesment methods and criteria linked to learning outcomes
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.
Language of instruction
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. 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).
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.
Specification of controlled education, way of implementation and compensation for absences
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.
Classification of course in study plans