Course detail
Mathematics 1 (G)
FAST-BAA008Acad. year: 2025/2026
Linear algebra (basics of matrix calculus, rank of a matrix, solution of linear systems by Gauss elimination method). Inverse matrix, determinants. Eigenvalues and eigenvectors of a matrix.
Geometrical vectors in three dimensional Euclidean space, operations with vectors. Applications of vector calculus in spherical trigonometry. Vector space, base, dimension, coordinates of a vector. Application of vector calculus in analytic geometry.Real function of one real variable, limit and continuity of a function (basic notions and properties), derivative of a function (geometrical and physical meaning, techniques of differentiation, basic theorems on derivatives, higher order derivatives, sketching the graph of a function, differentials of a function, Taylor expansion of a function).
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Department
Entry knowledge
Definition of a geometric vector and basics of 3D analytic geometry (parametric equations of a straight line, dot product of vectors and its applications to metric and positional problems). Identifying the the types and basic properties of conics, sketching graphs of conics).
Rules for evaluation and completion of the course
Aims
After passing the course students will have necessary skills for performing operations with vectors defined either generally or by their coordinates. Except this application of vectors in the spherical trigonometry will be deeply explained. The next outpiuts are: application of vector calculus in metrix and positional problems in analytical geometry, operations with matrices and solving systems of linear algebraic equations.
Bringing off basic differential calculus will permit successfully analyse problems of behavior of analytical curves.
Study aids
Prerequisites and corequisites
Basic literature
CHRASTINOVÁ, Veronika: Matematika I, Modul 3, Vektorová algebra a analytická geometrie, Stavební fakulta, Vysoké učení technické v Brně, Brno 2004. https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp
LARSON, Ron, HOSTETLER, Rober, EDWARDS Bruce: Calculus With Analytic Geometry, 8th edition, Brooks Cole, 2005. ISBN: 978-0618502981
NOVOTNÝ, Jiří: Matematika I, Modul 1, Základy lineární algebry, Akademické nakladatelství CERM, s.r.o., Brno 2004. ISBN: 978-80-7204-748-2
TRYHUK, Václav, DLOUHÝ, Oldřich: Matematika I, Modul GA01–M01, Vybrané části a aplikace vektorového počtu, Akademické nakladatelství CERM, s.r.o., Brno 2004. ISBN: 978-80-7204-526-6
Recommended reading
Classification of course in study plans
- Programme BPC-GK Bachelor's 1 year of study, winter semester, compulsory
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
- 1. Matrices, systems of linear algebraic equations, Gaussian elimination method.
- 2. Inverse matrix, determinants.
- 3. Geometrical vectors in three dimensional Euclidean space, operations with vectors.
- 4. Applications of vector calculus in spherical trigonometry.
- 5. Vector space, basis, dimension, coordinates of a vector.
- 6. Eigenvalues and eigenvectors of a matrix.
- 7. Application of vector calculus in analytic geometry.
- 8. Real function of one real variable, explicit and parametric expression of a function. Basic properties of functions. Composite fuction and inverse function. Elementary functions (including inverse trigonometric functions and hyperbolic functions).
- 9. Polynomials and rational functions.
- 10. Sequences and their limits, limit and continuity of a function.
- 11. Derivative of a function, its geometrical and physical meaning, derivation rules. Derivative of a composite function and of an inverse function. Derivatives of elementary functions.
- 12. Derivatives of higher order, geometrical meaning of first order and second order derivatives for sketching the graph of a function, l Hospital's rule, asymptotes.
- 13. Properties of functions continuous on an interval. Basic theorems of differential calculus (the Rolle and Lagrange theorems). Differential of a function. Taylor's theorem. Derivative of a function given in a parametric form.
Exercise
Teacher / Lecturer
Syllabus
- 1. Geometrical vectors in E3, operations with vectors.
- 2. Applications of vector calculus in spherical trigonometry.
- 3. Vector space, base, dimension, coordinates of a vector.
- 4. Application of vector calculus in analytic geometry.
- 5. Matrices, systems of linear algebraic equations, Gaussian elimination method.
- 6. Inverse matrix, determinants.
- 7. Eigenvalues and eigenvectors of a matrix.
- 8. Real function of a one real variable, explicit and parametric expression of a function. Basic properties of functions. Composite and inverse functions. Elementary functions.
- 9. Polynomials and rational functions.
- 10. Sequences and theirs limits, limit and continuity of a function.
- 11. Derivative of a function, its geometrical and physical meaning, derivation rules. Derivative of a composite function and of an inverse function. Derivatives of elementary functions.
- 12. Derivatives of higher order, geometrical meaning of first and second order derivatives for investigation of behavior of a function, l`Hospitals rule, asymptotes.
- 13. Properties of function, continuous on an interval. Basic theorems of differential calculus. Differential of a function. Taylor’s theorem. Derivative of a function given in a parametric form. Primitive function, Newtons integral, its properties and computation. Riemann’s integral. Integration methods for indefinite and definite integrals.