Linear Algebra and Differential Calculus

Lecturer:

Tutor:

Supervisor:

Department of Natural Sciences

Synopsis:

The subject is an introduction to differential calculus, linear algebra, and applications in analytical geometry .

Differential calculus: intro to sets of numbers (natural, rational, irrational), sequences of real numbers (bounded, unbounded, monotonous, convergent, divergent), real-valued functions of one independent real variable (injective, monotonous, continuous, differentiable), inverse functions, differential of a function, Taylor polynomials, number series.

Linear algebra: intro to solving linear algebraic systems of equations (LAES), Gaussian elimination algorithm for solving LAES, intro to vector spaces (linear combination of vectors, linear dependence/independence of a set of vectors, basis and dimension of a vector space), intro to matrix algebra (matrix operations, rank of a matrix, determinant and methods of calculation, matrix inverse, eigenvalues and eigenvectors of a matrix).

Requirements:

Assessment:

Maximum 3 absences during the semester for serious reasons like sickness, injury etc. (medical certificate required). Correct competion of the semester task and its in time deliveery.

Exam:

1. Assessment recorded in KOS and signed by respective teacher in student?s Index,

2. Minimum 50% evaluation at the exam test. Exam test comprises of 10 tasks, a single task evaluated max 10% each.

Evaluation scale: less than 50% - F, 50-59% - E, 60-69% - D, 70-79% - C, 80-89% - B, 90-100% - A.

Syllabus of lectures:

1. Sets of numbers, sequences of real numbers, limit of a sequence, properties of sequences (monotonous, bounded, convergent or divergent sequence), real-valued functions of one independent real variable, function properties (injective, bounded, unbounded, even, odd, periodic functions), operations with functions, compound functions, inverse function.

2. Review of elementary functions (polynomials, trigonometric functions, inverse trigonometric functions, exponential function, logarithmic function), limit of a function definition, techniques and rules for finding limits of a function, improper limits, limits in infinity, continuous functions, properties of continuous functions in a closed interval.

3. Asymptotes of a function -horizontal, vertical, oblique. Derivative of a function, rules and techniques for computing derivatives, derivative of a compound function, derivative of an inverse function, differential of a function and its application.

4. Tangent lines of a functions, L' Hospital rule, rules and techniques for computing higher order derivatives of a function, Taylor polynomial, Taylor's theorem in one real variable.

5. Local and global extremes of a function, monotonicity and the sign of the first derivative of a function, first and second derivatives use to find the graph of a given function.

6. Real number series, convergence criteria, sum of a series. (Test No. 1)

7. Systems of linear algebraic equations, Gaussian elimination algorithm for solving LAES,

8. Vector space - definition, examples, linear combination of vectors, linear dependence and independence of a family of vectors, linear span of a set of vectors, linear space and examples, basis and dimension of a linear space (or a subspace), dot product of two vectors.

9. Matrices, various types of matrices, rank of a matrix, operations with matrices, identity matrix, transpose of a matrix, matrix inverse, regular matrices, singular matrices.

10. Determinant of a square matrix, Sarrus? rule for calculating a determinant of a 3 by 3 matrix, Laplace rule, techniques for finding the inverse of a regular matrix.

11. Solvability of a system of linear algebraic equations, Frobenius theorem, structure of solution of a system of linear algebraic equations, methods for solving systems of linear equations, solving a system of linear algebraic equations with regular matrix - Cramer?s rule.

12. Eigenvalues and eigenvectors of square matrices.

13. Norm of a vector, angle of two vectors, dot product, vector product, and mixed (scalar triple) product and their application. (Test No. 2).

14. Selected tasks from analytical geometry in 2D and 3D Euclidian space.

Syllabus of tutorials:

Problems to be solved:

1. Properties of given sequences, limits of various sequences, convergence or divergence of a sequence, elementary functions and their properties (injective, non-injective, even, odd, periodical), domain, range and graph of some elementary function.

2. Operations with functions, limit of a given function, techniques and rules for finding limits of a function, improper limits, limits in infinity, continuity of a given function, continuous functions in a closed interval.

3. Asymptotes of a given function -horizontal, vertical, oblique. Practice techniques of differentiation of given functions derivative of a function, rules and techniques for computing derivatives, derivative of a compound function, derivative of an inverse function, differential of a function and its application.

4. Tangential lines for a given function, L' Hospital rule for finding limit of a function.

5. Local and global extremes of a function, monotonicity and the sign of the first derivative of a function, first and second derivatives use to find the graph of a given function. Taylor polynomial.

6. Analysis of properties of a given function, graph of the function.

7. Examples of series, geometrical series, convergence criteria, sum of a series.

8. Systems of linear algebraic equations, practice Gaussian elimination algorithm for solving LAES.

9. Vector space properties, linear dependence or independence of a family of vectors, linear space and examples, basis and dimension of a linear space (or a subspace) examples.

10. Dot product, operations with matrices, rank of a matrix, transpose of a matrix, matrix inverse, regular matrices, singular matrices. Gauss-Jordan elimination method for calculating inverse of a given matrix.

11. Sarrus rule for calculating a determinant of a 3 by 3 matrix, Laplace rule, techniques for finding the inverse of a regular matrix, solving matrix equations.

12. Application of Frobenius theorem, finding all solutions of a system of linear algebraic equations, Gausssian elimination method for solving systems of linear equations, solving a system of linear algebraic equations with regular matrix - Cramer?s rule.

13. Norm of a vector, angle of two vectors, dot product, vector product, and mixed (scalar triple) product and their application. (Test No. 2).

14. Selected tasks from analytical geometry in 2D and 3D Euclidian space.

Study Objective:

To gain necessary theoretical background of respected introductory fields of mathematics. By the end of the course students will be able to apply the knowledge in solving various practical examples and problems in the field of fundamental differential calculus and linear algebra.

Study materials:

[1] L. Gillman, R.H. McDowell: Calculus, Norton, New York, 1973

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Bakalářský studijní obor Biomedicínský technik - kombinovaný (compulsory course)