Linear Algebra via Exterior Products (original) (raw)

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.