Mathematik für Ingenieure I

Inhalte

Fundamentals: Kinds of mathematical statements and reasoning (direct proof, indirect proof, proof by complete induction), essential statements of combinatorics and sum formulas, set theory (relations and operations), definition and examples of mappings and functions, real numbers (working with inequalities and absolute values, infimum and supremum), b-adic expansions, complex numbers (Cartesian, polar, and Euler representation, number operations in these presentations, determination of roots). Analytic Geometry: Vectors in the plane and in space (representation, operations, scalar product, vector product, triple product), representation of lines (point-direction and two-point equation, distance formulas), planes (point-directions equation, three-points equation, Hesse normal form). Linear Algebra: Vectors and matrices (representation and operations, systems of linear equations (representation and solvability), Gauss algorithm, rank of a matrix, linear dependence and independence of vectors, representation of the solution set of a homogeneous and inhomogeneous system of linear equations by linearly independent solutions of the homogeneous system, LU factorization by Gauss algorithm and solution of systems of linear equations by that, determinant of a matrix (definition, computation via Gauss algorithm and Laplace expansion), inverse matrix (existence and computation via Gauss algorithm), orthogonal vectors and matrices (definitions, properties, Gram-Schmidt procedure), QR factorization of a matrix and the solution of systems of linear equations by that, linear mappings (definition, orthogonal mappings and their geometrical properties), eigenvalues and eigenvectors (definition, computation, results on existence of linear independent eigenvectors), diagonalization of matrices (principal axes transformation and its application to quadratic equations), definiteness properties of matrices (definition and verification via computation of eigenvalues).