Intro
Foreword to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
1 Speech, Symbols and Sets
1.1 Speech Patterns and Symbols in Mathematics
1.1.1 Junctors
1.1.2 Quantifiers
1.2 Summation and Product Symbol
1.3 Powers and Roots
1.4 Symbols of Set Theory
1.5 Exercises
2 The Natural Numbers, Integers and Rational Numbers
2.1 The Natural Numbers
2.2 The Integers
2.3 The Rational Numbers
2.4 Exercises
3 The Real Numbers
3.1 Basics
3.2 Real Intervals
3.3 The Absolute Value of a Real Number

3.4 n-th Roots
3.5 Solving Equations and Inequalities
3.6 Maximum, Minimum, Supremum and Infimum
3.7 Exercises
4 Machine Numbers
4.1 b-adic Representation of Real Numbers
4.2 Floating Point Numbers
4.2.1 Machine Numbers
4.2.2 Machine Epsilon, Rounding and Floating Point Arithmetic
4.2.3 Loss of Significance
4.3 Exercises
5 Polynomials
5.1 Polynomials: Multiplication and Division
5.2 Factorization of Polynomials
5.3 Evaluating Polynomials
5.4 Partial Fraction Decomposition
5.5 Exercises
6 Trigonometric Functions
6.1 Sine and Cosine

6.2 Tangent and Cotangent
6.3 The Inverse Functions of the Trigonometric Functions
6.4 Exercises
7 Complex Numbers: Cartesian Coordinates
7.1 Construction of C
7.2 The Imaginary Unit and Other Terms
7.3 The Fundamental Theorem of Algebra
7.4 Exercises
8 Complex Numbers: Polar Coordinates
8.1 The Polar Representation
8.2 Applications of the Polar Representation
8.3 Exercises
9 Linear Systems of Equations
9.1 The Gaussian Elimination Method
9.2 The Rank of a Matrix
9.3 Homogeneous Linear Systems of Equations
9.4 Exercises

10 Calculating with Matrices
10.1 Definition of Matrices and Some Special Matrices
10.2 Arithmetic Operations
10.3 Inverting Matrices
10.4 Calculation Rules
10.5 Exercises
11 LR-Decomposition of a Matrix
11.1 Motivation
11.2 The LR-Decomposition: Simplified Variant
11.3 The LR-Decomposition: General Variant
11.4 The LR-Decomposition-with Column Pivot Search
11.5 Exercises
12 The Determinant
12.1 Definition of the Determinant
12.2 Calculation of the Determinant
12.3 Applications of the Determinant
12.4 Exercises
13 Vector Spaces

13.1 Definition and Important Examples
13.2 Subspaces
13.3 Exercises
14 Generating Systems and Linear (In)Dependence
14.1 Linear Combinations
14.2 The Span of X
14.3 Linear (In)Dependence
14.4 Exercises
15 Bases of Vector Spaces
15.1 Bases
15.2 Applications to Matrices and Systems of Linear Equations
15.3 Exercises
16 Orthogonality I
16.1 Scalar Products
16.2 Length, Distance, Angle and Orthogonality
16.3 Orthonormal Bases
16.4 Orthogonal Decomposition and Linear Combination with Respect to an ONB
16.5 Orthogonal Matrices
16.6 Exercises