Code: F7ABBLAD Linear Algebra and Differential Calculus
Lecturer: Ing. Tomáš Parkman Ph.D. Weekly load: 2P+4C Completion: A, EX
Department: 17101 Credits: 6 Semester: W
Description:
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.
Contents:
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. L?Hospital?s 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. Taylor?s polynomial. Taylor?s series. Concrete examples: Taylor?s polynomials and
Taylor?s 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. Cramer?s rule. Eigenvalues and eigenvectors of squate matrices.
Seminar contents:
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 l?Hospital?s 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 Cramer?s 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.
Recommended literature:
[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/
Keywords:
Linear algebra, matrices, Gaussian elimination, differential calculus, limit of a function, derivative of a function, investigation of functions, Taylor polynomial, number series

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