A concise text on advanced linear algebra
type: Item
instance of: scholarly article
zbMATH Open document ID: 1311.15001
DOI: 10.1017/CBO9781316103845
author: Yisong Yang
publication date: 2014-10-22T00:00:00Z
Mathematics Subject Classification ID: 15-01
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A16
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 6359813
zbMATH Keywords: textbook
zbMATH Keywords: matrix exponential
zbMATH Keywords: vector space
zbMATH Keywords: dual space
zbMATH Keywords: direct sum
zbMATH Keywords: quotient space
zbMATH Keywords: vector norm
zbMATH Keywords: linear mapping
zbMATH Keywords: change of bases
zbMATH Keywords: adjoint mapping
zbMATH Keywords: quotient mapping
zbMATH Keywords: determinant
zbMATH Keywords: Cayley-Hamilton theorem
zbMATH Keywords: scalar product
zbMATH Keywords: quadratic form
zbMATH Keywords: positive definite form
zbMATH Keywords: diagonalization
zbMATH Keywords: normal operator
zbMATH Keywords: Jordan canonical form
zbMATH Keywords: Schur triangularization
zbMATH Keywords: Perron-Frobenius theorem
zbMATH Keywords: positive matrices
zbMATH Keywords: Markov matrix
zbMATH Keywords: quantum mechanics
MaRDI profile type: Publication
full work available at URL: cbo9781316103845
OpenAlex ID: W4301011744
title: A concise text on advanced linear algebra
review text: 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 looki…
reviewed by: John D. Dixon
description: scientific article; zbMATH DE number 6359813