Linear Algebra and Differential Calculus

Lecturer:

Eva Feuerstein (gar.)

Tutor:

Eva Feuerstein (gar.)

Supervisor:

Department of Natural Sciences

Synopsis:

The course is an introduction to differential calculus and linear algebra.

Differential calculus:sequences and their limits, functions of one real variable, their limits,

continuity, derivatives, local and global extremes of a function, investigations of functions. Taylor -polynomial.

Linear algebra: systems of linear algebraic equations and their solutions, Gaussian elimination, matrices, rank of a matrix, operations with matrices, inverse matrix, determinant and its calculation, eigenvalues and eigenvectors of matrices.

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

14. Various examples of exam tests solving.

Study Objective:

The course is an introduction into linear algebra and calculus of one variable.

Linear algebra part consists of: systems of linear equations and their solutions, Gauss elimination, matrices, rank of a matrix, operations with matrices, inverse matrix, determinant and its calculation, eigenvalues and eigenvectors of matrices.

Differential calculus consists of: sequences and their limits. Functions of one real variable, their limits,

continuousness, derivatives. Local and absolute extremes of a function of one variable, investigations of functions. Taylor-polynome.

Study materials:

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

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans:

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