Linear algebra and differential calculus

Lecturer:

Tutor:

Supervisor:

Department of Natural Sciences

Synopsis:

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,

continuity, derivatives. Local and absolute extrema of a function of one variable, investigations of functions. Taylor-polynomial.

Requirements:

Credit condition - 70% presence, successful written test in 6. and 13. week. It is necessary to gain at least one half of maximum number of points.

Theme of 1. test: limits, asymptotes, tangents, local extrema.

Theme of 2. test: linear independence of vectors, inverse matrix, matrix equation, determinant.

Examine condition - credit in index and in KOS,

test - 6 examples, each 10 points plus 2 theoretical questions, each 20 points. As a whole 100 points is the best result and 50 points is a minimum for to be still successful.

Syllabus of lectures:

1. Number sets, sequences, limit of sequence, convergence of sequence. Functions of one real variable, properties, operations with functions. composed function, inverse function.

2. Limit and continuity of function, rules for calculation of limits, infinite limits, right-hand, left-hand limits.

3. Asymptotes, derivative, rules for calculation, derivative of composite function, inverse function, higher order derivative.

4. Differential of function and its application, properties of a function continuous on a closed interval, L'Hospital rule, implicit functions.

5. Local and global extrema, graph of function.

6. Taylor polynomial, number series, criteria of convergence, sum of series.

7. Gauss elimination method of solution of linear algebraic equation system (LAES). Vector spaces, subspaces, their properties.

8. Linear combinations of vectors, linear (in)dependence of vector system, base and dimension, scalar product.

9. Matrices, rank of matrix, product of matrices, inverse matrix, regular and singular matrices.

10. Permutation, determinant of a square matrix, Sarrus rule, calculation of inverse matrix.

11. Solution of LAES , Frobenius theorem, equivalent systems, structure of general solution of LAES, system with regular matrix, Cramer rule.

12. Coordinates of a vector in given baze. Eigen values and eigen vectors of a matrix. Angle of two vectors, scalar and vector product, application.

13. Some notes to analytical geometry of E2, E3 spaces, conics.

14. Recapitulation.

Syllabus of tutorials:

1. Sequences, limits, elementary functions.

2. Operations with functions, properties, limit of function, continuity.

3. Asymptotes, inverse function, derivative of function.

4. Intervals of monotony, L'Hospital rule for limits.

5. Investigation of function, local and global extrema.

6. Taylor polynomial, number series, convergence. Test 1.

7. Gauss elimination, vector spaces.

8. Linear (in)dependence of vectors, base, dimension.

9. Matrices, inverse matrix, product of matrices.

10. Calculation of determinant, Sarrus rule.

11. LAES solution.

12. Coordinates of vector in given base, eigenvalue and eigen vectors of a square matrix.

13. Analytical geometry in a plane and in a space. Test 2.

14. Revision, credit.

Study Objective:

The goal of study is to get a notion about base of differential calculus and linear algebra and some applications of theory.

Study materials:

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 - prezenční v angličtině (compulsory course)