Linear Algebra and Differential Calculus

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:

Tomáš Parkman

Lecturer:

Jiří Neustupa, Tomáš Parkman

Tutor:

Lukáš Liebzeit, Jiří Neustupa, Tomáš Parkman

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:

Form of verification of learning outcomes and other requirements for the student:

A - Compulsory attendance at exercises, absences must be duly excused in advance and subsequently documented e.g. by a medical certificate. Maximum of three properly excused absences. Active participation in the exercises is evaluated (maximum 5 points per semester, and these are added to the exam grade).

Attendance at lectures is not compulsory, but if a student fails to attend a lecture, he/she is obliged to supplement the material by self-study and must come prepared to the exercise.

B - Knowledge in the scope of individual topics of lectures is tested by two semester tests, which students take together in a single term in the middle and at the end of the semester according to the teaching schedule of the subject for the given academic year. Calculators certified for the mathematics graduation exam (i.e., non-programmable, without integrals and equation solving) and a list of formulas that will be included in the test assignment are allowed in the tests.

A condition for the award of credit is the fulfilment of point A and the achievement of at least 50% of the points from both semester tests (each test has a maximum of 40 points, the minimum for successful completion is 40 points in total).

A condition for admission to the examination is the credit entered in KOS. The exam is written only, lasts 120 minutes, calculators and formula lists are allowed, the same as for the mid-semester tests. The exam contains mostly numerical examples supplemented by theoretical subquestions within the of the material covered in lectures. The examination test consists of numerical problems from the material covered in the lectures and exercises supplemented by theoretical sub-questions. The maximum score is 75 points, for successful completion of the examination test, the student must obtain at least half of the points (i.e. 37.5 points). The points from both semester tests will be added to the exam score as follows: points above the mandatory 50% divided by 2 (max. 20 points) and points for activity in the exercises (max. 5 points). The total number of points is therefore (75+20+5) 100

Course grade: A: 100-90, B: 89-80, C: 79-70, D: 69-60, E: 59-50, 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. LHospitals 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. Taylors polynomial. Taylors series. Concrete examples: Taylors polynomials and

Taylors 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. Cramers 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 lHospitals 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 Cramers 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, 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/

Note:

Further information:

https://predmety.fbmi.cvut.cz/en/17ABBLAD

Time-table for winter semester 2025/2026:

	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	roomKL:B-307 Neustupa J. 14:00–15:50 (lecture parallel1) Kladno FBMI roomKL:B-326 Liebzeit L. 16:00–19:50 (lecture parallel1 parallel nr.1) Kladno FBMI
Thu
Fri

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

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