Winitzki S. Linear Algebra via Exterior Products

Ver 1.3. — Github, 2020. — 285 p.This book is an undergraduate-level introduction to the coordinate-free approach in basic finite-dimensional linear algebra. The reader should be already exposed to the elementary array-based formalism of vector and matrix calculations. Throughout this book, extensive use is made of the exterior (anti-commutative, “wedge”) product of vectors. The coordinate-free formalism and the exterior product, while somewhat more abstract, provide a deeper understanding of the classical results in linear algebra. The standard properties of determinants, the Pythagoras theorem for multidimensional volumes, the formulas of Jacobi and Liouville, the Cayley-Hamilton theorem, properties of Pfaffians, the Jordan canonical form, as well as some generalizations of these results are derived without cumbersome matrix calculations. For the benefit of students, every result is logically motivated and discussed. Exercises with some hints are provided. This book is largely pedagogical, meaning that the results are long known, and the emphasis is on a clear and self-contained, logically motivated presentation aimed at students. Therefore, some exercises with hints and partial solutions are included, but not references to literature. Sections marked with a star ∗ are not especially difficult but contain material that may be skipped at first reading (exercises marked with a star are more difficult).
In the first 10 years since book is available for free, many readers sent corrections to the author. Version 1.3 (prepared in June 2020) incorporates these corrections.Introduction and summary.
Linear algebra without coordinates.
Exterior product.
Basic applications.
Advanced applications.
Scalar product.
A Complex numbers.
B Permutations.
C Matrices.
D Distribution of this text.True PDF