Code: 17ABBLAD Linear Algebra and Differential Calculus
Lecturer: RNDr. Eva Feuerstein Ph.D. Weekly load: 2P+2C Completion: A, EX
Department: 17101 Credits: 4 Semester: W
Description:
The course is introduction to differential calculus and linear algebra.
Differential calculus - sets of numbers, sequences of real numbers, real functions (function properties, limits, continuity and derivative of a function investigation of function behavior), Taylor's formula, real number series.
Linear algebra - vector spaces, matrices and determinants, systems of linear algebraic equations (solvability and solution), eigenvalues and eigenvectors of matrices, applications.
Contents:
1. Number sets, sequences, limit of sequence, convergence of sequence. Functions of one real variable, properties, operations with functions. composed function, inverse function.
2. Limit and continuity of function, rules for calculation of limits, infinite limits, right-hand, left-hand limits.
3. Asymptotes, derivative, rules for calculation, derivative of composite function, inverse function, higher order derivative.
4. Differential of function and its application, properties of a function continuous on a closed interval, L'Hospital rule, implicit functions.
5. Local and global extrema, graph of function.
6. Taylor polynomial, number series, criteria of convergence, sum of series.
7. Gauss elimination method of solution of linear algebraic equation system (LAES). Vector spaces, subspaces, their properties.
8. Linear combinations of vectors, linear (in)dependence of vector system, base and dimension, scalar product.
9. Matrices, rank of matrix, product of matrices, inverse matrix, regular and singular matrices.
10. Permutation, determinant of a square matrix, Sarrus rule, calculation of inverse matrix.
11. Solution of LAES , Frobenius theorem, equivalent systems, structure of general solution of LAES, system with regular matrix, Cramer rule.
12. Coordinates of a vector in given baze. Eigen values and eigen vectors of a matrix. Angle of two vectors, scalar and vector product, application.
13. Some notes to analytical geometry of E2, E3 spaces, conics.
14. Recapitulation.
Recommended literature:
[1] Neustupa, J. : Mathematics 1, textbook, ed. ČVUT, 2004
[2] Bubeník F.: Problems to Mathematics for Engineers, textbook, ed. ČVUT, 2007
[3] Neustupa, J., Kračmar s.: Sbírka příkladů z matematiky I, skriptum ČVUT 2003
[4] Tkadlec, J.: Diferenciální a integrální počet funkcí jedné proměnné, skriptum ČVUT, 2004
[5] Stewart, J.: Calculus, 2012 Brooks/Cole Cengage Learning, ISBN-13: 978-0-538-49884-5
[4] http://mathonline.fme.vutbr.cz/?server=2
[5] http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/
[6]http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/
[7] http://www.studopory.vsb.cz
Keywords:
Linear algebra, matrices, Gaussian elimination, differential calculus, derivative, investigation of functions

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