 CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2023/2024

# Linear Algebra and Differential Calculus

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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:

Assessment

8 mini-tests graded with max 5 points (PTS), MP- total number of points achieved.

Minimum 9 PTS per test required, but the sum of PTS achieved in both mid-term tests must be at least 20 PTS. BP - total number of points achieved in both mid-term tests.

Points transferred for the exam: GP =MP/8+BP/4 ranging from 5 to 15 PTS.

1st Midterm Test - 22nd November, 2016, 8pm, NTK

Themes: sequences, real function domain, asymptotes, tangent lines, local extremes, monotonicity, concavity, inflection points.

2nd Midterm Test - 20th December, 2016, 8PM, NTK

Themes: linear dependence/independence of a set of vectors, determinants, matrix equation solution, linear systems solvability and solution.

Exam

It is forbidden to use a calculator or a mobile telephone or another electronic device during the exam.

Exam written test

5 multiple choice tests with 1 grading point each (max 5 points)

5 multiple choice tests with 2 grading points each (max 10 points)

Transferred points GP (from 5 up to maximum 15 points)

Grading and evaluation: 100-90 A, 89-80 B, 79-70 C, 69-60 D, 59-50 E, less than 50 F.

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, limits, elementary functions.

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

3. Asymptotes, inverse function, derivative of function.

4. Intervals of monotony, L'Hospital rule for limits.

5. Investigation of function, local and global extrema.

6. Taylor polynomial, number series, convergence. Test 1.

7. Gauss elimination, vector spaces.

8. Linear (in)dependence of vectors, base, dimension.

9. Matrices, inverse matrix, product of matrices.

10. Calculation of determinant, Sarrus rule.

11. LAES solution.

12. Coordinates of vector in given base, eigenvalue and eigen vectors of a square matrix.

13. Analytical geometry in a plane and in a space. Test 2.

14. Revision, credit.

Study Objective:

The goal of the study is to learn fundamental topics of differential calculus and linear algebra with some applications, which are topics of study in the subsequent specific subjects of the program.

Study materials:

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

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

 Neustupa, J., Kračmar s.: Sbírka příkladů z matematiky I, skriptum ČVUT 2003

 Tkadlec, J.: Diferenciální a integrální počet funkcí jedné proměnné, skriptum ČVUT, 2004

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

Note:
Further information:
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2023-10-04
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