# F7ABBLAD - Linear Algebra and Differential Calculus

Code Completion Credits Range Language
Garant předmětu:
Petr Maršálek
Lecturer:
Jiří Neustupa
Tutor:
Petr Maršálek, Jiří Neustupa
Supervisor:
Department of Natural Sciences
Synopsis:

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.

Requirements:
Syllabus of lectures:

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.

Syllabus of tutorials:

1. Sequences, properties of sequences, limit of a sequence and calculation, elementary functions.

2. Operations with functions, properties, limit of a function, continuity.

3. Asymptotes of a function, inverse function, derivative of a function. Differential and its application for calculating approximate value of a function

4. Function and its intervals of monotony, L'Hospital's rule for calculating limits.

5. Investigation of function behavior, local and global extrema, convexity concavity, points of inflexion.

6. Taylor polynomial, number series, convergence/divergence of a series.

7. Gaussian elimination, linear independence/dependence of a set of vectors.

8. Vector spaces(VS) and subspaces(VSS), base and dimension of VS,VSS.

9. Matrices, algebraic operations on matrices, inverse of a matrix.

10. Calculation of determinant, Sarrus's rule.

11. Solvability and solution of a Linear Algebraic Equations system.

12. Coordinates of vector wrt given base, eigenvalues and eigenvectors of a square matrix.

13. Analytical geometry in a plane and in a space.

14. Application of determinants - Conic Sections and Quadrics classification.

Study Objective:

The goal of the study is to learn fundamental topics of differential calculus and linear algebra and gain skills in solving relevant examples and real life problems corresponding to key subjects of the study program.

Study materials:

 Neustupa, J. : Mathematics 1, textbook, ed. ČVUT, 2004

 Bubeník F.: Problems to Mathematics for Engineers, textbook, ed. ČVUT, 2007

 Stewart, J.: Calculus, 2012 Brooks/Cole Cengage Learning, ISBN-13: 978-0-538-49884-5

Note:
The course is a part of the following study plans:

Lectures:
AttachmentSize F7ABBLAD_L1 Number sets, Sequences - Feuerstein AY 2021/22402.13 KB F7ABBLAD_L2 Intro to Matrix Calculus - Feuerstein AY 2021/22258.11 KB F7ABBLAD_L2 BAMBOO NOTES Feuerstein AY 2021/2217.9 MB F7ABBLAD_L3 Intro to Functions - Feuerstein AY 2021/22778.85 KB F7ABBLAD_L4 Functions - Limits, Continuity, Asymptotes - Feuerstein AY 2021/22605.84 KB F7ABBLAD_L5 Functions - DERIVATIVE, TANGENT and NORMAL LINES, L'HOSPITAL RULE - Feuerstein AY 2021/22342.89 KB F7ABBLAD_L6 Functions - DERIVATIVE APPLICATIONS 1 - Feuerstein AY 2021/22417.11 KB F7ABBLAD_L6 B Functions - DERIVATIVES, FURTHER APPLICATIONS - Feuerstein AY 2021/22270.29 KB F7ABBLAD_L7 SERIES, CONVERGENCE, DIVERGENCE, SUM, CONVERGENCE CRITERIA - Feuerstein AY 2021/22313.17 KB Lecture in English in the 1st week - J.Neustupa74.75 KB Lecture in English in the 2nd week - J.Neustupa239.63 KB Lecture in English in the 6th week - J.Neustupa194.08 KB Lecture in English in the 7th week - J.Neustupa262.02 KB Lecture in English in the 8th week - J.Neustupa1.37 MB Lecture in English in the 9th week - J.Neustupa142 KB Lecture in English in the 10th week - J.Neustupa139.06 KB Lecture in English in the 11th week - J.Neustupa 138.23 KB

Exercises:
AttachmentSize F7ABBLAD_S1 Number sets, Sequences - Feuerstein AY 2021/22 update809.85 KB F7ABBLAD_S2 Intro to Matrix Calculus - Feuerstein AY 2021/22145.57 KB F7ABBLAD_S2 BAMBOO NOTES Feuerstein AY 2021/226.16 MB F7ABBLAD_S3 Functions, properties - Feuerstein AY 2021/22196.04 KB F7ABBLAD_S4 Function limits and continuity, asymptotes - Feuerstein AY 2021/22200.22 KB F7ABBLAD_S5 Function derivative, tangent and normal lines, L'Hospital rule - Feuerstein AY 2021/22158.18 KB F7ABBLAD_S5-6 Function derivatives properties and further Applications - Feuerstein AY 2021/22233.77 KB F7ABBLAD_L&S BAMBOO NOTES 18.10.21 Feuerstein AY 2021/2222.36 MB SCHEDULE - F7ABBLAD-LECTURES-2021-22109.96 KB SCHEDULE - F7ABBLAD-SEMINARS AND LABS - 2021-22203.02 KB ASSESSMENT AND EXAM CONDITIONS F7ABBLAD & 17ABBLAD AY 2021/22153.27 KB TOPICS for EXAMS F7ABBLAD & 17ABBLAD AY 2021/2296.51 KB SCHEDULE - F7ABBLAD-SEMINARS AND LABS - 2021-22191.67 KB