Linear algebra via exterior products
by Sergei Winitzki
Publisher: Ludwig-Maximilians University 2009
Number of pages: 82
A pedagogical 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. In this book, the author makes extensive use of the exterior product of vectors. He shows how the standard properties of determinants, the Liouville formula, the Hamilton-Cayley theorem, and Pfaffians, as well as some results concerning eigenspace projectors can be derived without cumbersome matrix calculations.
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by Peter Petersen - UCLA
This book covers the aspects of linear algebra that are included in most advanced undergraduate texts: complex vectors spaces, complex inner products, spectral theorem for normal operators, dual spaces, quotient spaces, the minimal polynomial, etc.
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This book is written as a text for a second semester of linear algebra at the senior or first-year-graduate level. It is assumed that you already have successfully completed a first course in linear algebra and a first course in abstract algebra.
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From the tabble of contents: Basic Concepts and Notation; Matrix Multiplication; Operations and Properties; Matrix Calculus (Gradients and Hessians of Quadratic and Linear Functions, Least Squares, Eigenvalues as Optimization, etc.).