Linear Algebra and Differential Calculus
Code  Completion  Credits  Range  Language 

F7ABBLAD  Z,ZK  6  2P+4C  English 
 Relations:
 In order to register for the course F7ABBITP, the student must have successfully completed or received credit for and not exhausted all examination dates for the course F7ABBLAD.
 The course F7ABBPMS can be graded only after the course F7ABBLAD has been successfully completed.
 Course guarantor:
 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:

The form of the method of verification of study results and other requirements for the student
A  Compulsory participation in exercises, absences must be duly excused in advance and subsequently documented, e.g. medication. confirmation. Depending on the situation, they are either replaced in the exercises in the same week, or in the form of extraordinary tasks, which the student properly prepares and submits to his teacher on the agreed date.
Attendance at lectures is not compulsory, however, if the student does not attend the lecture, he is obliged to supplement the material covered by selfstudy and must come prepared to the exercises.
B  Knowledge in the scope of individual topics of the lectures is checked by two halfsemester tests, which students take at the same time in the middle and at the end of the semester according to the subject teaching schedule for the given academic year. Calculators with a certificate for the DT matriculation exam in mathematics (i.e. nonprogrammable, without integrals and solving equations) and a list of formulas that will be part of the test are allowed during the tests.
The condition for granting credit is the fulfillment of point A and the gain of at least 50% points from each halfsemester test (each test max. 50 points, the minimum is 25 points).
A condition for admission to the exam is a credit registered in KOS. The exam is written only, lasts 120 minutes, calculators and lists of formulas are allowed, the same as for the halfsemester tests. The exam mainly contains numerical examples supplemented by theoretical subquestions in the scope of the material covered. Points from both halfsemester tests will be added to the exam test score (75 points) as follows: score above the mandatory 50% divided by 2. The total number of points is therefore 100.
Course grading: A: 10090, B: 8980, C: 7970, D: 6960, E: 5950, F: less than 50.
 Syllabus of lectures:

1. Number sets, sequences of numbers, basic notions and properties of the sequences,
limit of a sequence.
2. More on the set of complex numbers, operations with complex numbers. Series
of real and complex numbers, sum of a series, comparison test for convergence.
Power series.
3. Real function of one real variable, basic notions, operations with functions, composite and inverse function, survey of elementary functions.
4. Limit of a function, basic properties. Improper limits and limits in improper
points. Continuity of a function at a point and in an interval, properties of continuous functions.
5. Derivative of a function, geometrical and physical meaning, basic properties and
formulas for derivatives of a sum, difference, product and quotient of two functions. Derivative of a composite and inverse function. Derivatives of elementary functions.
6. L’Hospital’s rule. Higher order derivatives. Investigation of local and global extremes of functions by means of the derivative.
7. Vertical and slant asymptotes of the graph of a function. Behavior of a function.
Differential of a function.
8. Taylor’s polynomial. Taylor’s series. Concrete examples: Taylor’s polynomials and
Taylor’s series of the exponential function and the functions sin x and cos x.
9. Vector space. Linear combination of vectors. Linear dependence and independence of vectors. Basis and dimension of a vector space.
10. Subspace of a vector space. Linear hull of a group of vectors. Matrices, types of
matrices, operations with matrices. Rank of a matrix, finding the rank.
11. A square matrix, identity matrix, inverse matrix, regular and singular matrices.
Determinant of a square matrix, methods of evaluation.
12. Relation between the determinant and the existence of an inverse matrix. Methods of evaluation of the inverse matrix. System of linear algebraic equations,
homogeneous and inhomogeneous system.
13. Structure of the set of all solutions of the hoimogeneous and inhomogeneous
system of linear algebraic equations. Gauss elimination method.
14. Frobenius’ theorem. Cramer’s rule. Eigenvalues and eigenvectors of squate matrices.
 Syllabus of tutorials:

Exercises outline
1. Testing secondary (high) school mathematics (this is not graded towards semester
evaluation). Repeating selected parts of high school mathematics.
2. DIFFERENTIAL CALCULUS part. Number sets. Number sequences, proper
and improper limits of sequences.
3. Number series, convergence criteria, sums of series.
4. Elementary functions, their properties. Surveing function properties and drawing
graphs. Composite, inverse, even and odd, continuous, piecewise continuous, discontinuous, and other types of function.
5. Function limit. Calculations of different types of limits. Existence, improper and
proper points and limits.
6. Calculations of derivatives of specific functions, application of formulas to calculate derivatives of elementary functions, differentiating sum, product and ratio of functions. Differentiating composite functions.
7. Evaluating limits using l’Hospital’s rule. Higher order derivatives. Calculating
function local and global extremes using derivatives. Convex and concave functions.
8. Tangent and normal of the function graph. Asymptotes of the graph. Investigating
function behavior. Taylor polynomials of selected functions.
9. LINEAR ALGEBRA part. Linear dependence and independence of vectors. Vector space basis. Writing vector with respect to various bases. Dimension of vector space and its subspaces and subsets.
10. Examples of subspaces. Matrix operations. Rank of matrix by the Gauss algorithm. Addition, subtraction and multiplication of square matrices, zero and unit matrix and other algebraic properties of matrices.
11. Calculations of determinants of square matrices, existence and calculation of inverse matrices using the determinant and alternatively using the Gauss algorithm.
12. System of linear algebraic equations is solved by Gauss elimination algorithm.
13. System of linear algebraic equations is solved by Cramer’s rule. Using Frobenius
theorem. Calculating square matrix eigenvalues and eigenvectors.
14. Points, vectors and lines in E3. Scalar and vector product of two vectors, angle of
two vectors. The relative position of the point and the line and other stuctures
in E3, their distance.
 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:

[1] Neustupa, J. : Mathematics 1, textbook, ed. ČVUT, 2004
[2] Bubeník F.: Problems to Mathematics for Engineers, textbook, ed. ČVUT, 2007
[3] Stewart, J.: Calculus, 2012 Brooks/Cole Cengage Learning, ISBN13: 9780538498845
[4] http://mathonline.fme.vutbr.cz/?server=2
[5] http://ocw.mit.edu/courses/mathematics/1801scsinglevariablecalculusfall2010/
[6]http://ocw.mit.edu/courses/mathematics/1806sclinearalgebrafall2011/
 Note:
 Further information:
 https://predmety.fbmi.cvut.cz/en/17ABBLAD
 Timetable for winter semester 2024/2025:

06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon Tue Wed Thu Fri  Timetable for summer semester 2024/2025:
 Timetable is not available yet
 The course is a part of the following study plans:

 Prospectus  bakalářský (!)
 Biomedical Technology (compulsory course)
 Biomedical Technology (compulsory course)