A concise text on advanced linear algebra
This book is a text in linear algebra for senior undergraduates and beginning graduate students in mathematics, science and engineering with an emphasis on the analytic aspects of the subject. The core of the book is the first seven chapters: 1. Vector spaces (a brief review of basic facts about general finite-dimensional vector spaces, followed by sections on dual spaces, direct sums, quotient spaces and vector norms); 2. Linear mappings (linear mappings and their matrices, change of bases, adjoint mappings, quotient mappings, norms on linear mappings and the series for \(e^{A}\)); 3. Determinants (definition via cofactors, properties and the Cayley-Hamilton theorem); 4. Scalar products (bilinear symmetric functions, positive definite scalar products over \(\mathbb{R}\), orthogonal resolution of vectors, isometric mappings); 5. Real quadratic forms and self-adjoint mappings (bilinear and quadratic forms, positive definite forms, positive definite matrices, diagonalization of pairs of commutative self-adjoint operators and normal operators); 6. Complex quadratic forms and self-adjoint mappings (extension of Chapter 5 to the complex case); 7. Jordan decomposition (Jordan canonical form). The last two chapters deal with special topics. Chapter 8 discusses Schur triangularization over an orthonormal basis, classification of skew-symmetric bilinear forms, the Perron-Frobenius theorem for positive matrices and limit properties of powers of a Markov matrix. Chapter 9 is an interesting outline of how basic results in quantum mechanics can be derived from a few fundamental postulate using linear algebra and elementary calculus. There are very few examples in the text, but the book includes good (sometimes challenging) exercises and worked-out solutions to selected exercises at the end of the book.NEWLINENEWLINEIt is assumed that a reader will have a strong background in undergraduate analysis and be comfortable with working in an abstract algebraic setting. If you are looking for a text for students with this background, then this book may be an interesting possibility.
scientific article; zbMATH DE number 6359813