This book is based on a course for first-semester science students, held by the second author at the University of Zurich several times. Its goal is threefold: to have students learn a minimal working knowledge of linear algebra, acquire some computational skills, and familiarize them with mathematical language to make mathematical literature more accessible. Therefore, we give precise definitions, introduce helpful notations, and state any results carefully worded. We provide no proofs of these results but typically illustrate them with numerous examples. Additionally, for better understanding, we often give supporting arguments for why they are valid.
Part I Systems of linear equations.- 1 Introduction.- 2 Systems with two
equations and two unknowns.- 3 Gaussian elimination.- Part II Matrices and
related topics.- 4 Basic operations.- 5 Linear dependence, bases,
coordinates.- 6 Determinants.- Part III Complex numbers.- 7 Complex numbers:
definition and operations.- 8 The Fundamental Theorem of Algebra.- 9 Linear
systems with complex coefficients.- Part IV Vector spaces and linear maps.-
10 Vector spaces and their linear subspaces.- 11 Linear maps.- 12 Inner
products on K-vector spaces.- Part V Eigenvalues and eigenvectors.- 13
Eigenvalues and eigenvectors of Clinear maps.- 14 Eigenvalues and
eigenvectors of R-linear maps.- 15 Quadratic forms on Rn.- Part VI
Differential equations.- 16 Introduction.- 17 Linear ODEs with constant
coefficients of first order.- 18 Linear ODEs with constant coefficients of
higher order.- Appendix A Solutions.
Manuel Benz is a high school teacher working in Zurich. After his studies in theoretical particle physics and mathematics, he taught, together with Thomas Kappeler, several courses at the University of Zurich. The courses' goal: To find a bridge between high school and university mathematics and to motivate young students to pursue their studies in mathematics. Thomas Kappeler was an Emeritus Professor at the University of Zurich. He started his academic career with a thesis on bilinear integrals, was a visiting professor at four universities in the United States and following a professorship at the Ohio State University, he was appointed professor at the University of Zurich. His research focused, among others, on global analysis and dynamical systems. In his lectures, he took great care to present the topics with precision and clarity.
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