Linear Algebra In Context
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Linear Algebra in Context
This text combines a compact linear algebra course with a serious dip into various physical applications. It may be used as a primary text for a course in linear algebra or as a supplementary text for courses in applied math, scientific computation, mathematical physics, or engineering. The text is divided into two parts. Part 1 comprises a fairly standard presentation of linear algebra. Chapters 1–3 contain the core mathematical concepts typical for an introductory course while Chapter 4 contains numerous "short" applications. Chapter 5 is a repository of standard facts about matrix factorization and quadratic forms together with the "connective tissue" of topics needed for a coherent discussion, including the singular value decomposition, the Jordan normal form, Sylvester's law of inertia and the Witt theorems. Part I contains around 300 exercises, found throughout the text, and are an integral part of the presentation. Part 2 features deeper applications. Each of these "large" applications require no more than linear algebra to discuss, though the style and arrangement of results would be challenging to a beginning student and more appropriate for a second or later course. Chapter 6 provides an introduction to the discrete Fourier transform, including the fast Fourier algorithm. Chapter 7 is a thorough introduction to isometries and some of the classical groups, and how these groups have come to be important in physics. Chapter 8 is a fairly detailed look at real algebras and completes a presentation of the classical Lie groups and algebras. Chapter 9 is a careful discussion of tensors on a finite-dimensional vector space, finishing with the Hodge Star operator and the Grassmann algebra. Finally, Chapter 10 gives an introduction to classical mechanics including Noether's first theorem and emphasizes how the classical Lie groups, discussed in earlier chapters, become important in this setting. The Chapters of Part 2 are intended to give a sense of the ubiquity, of the indispensable utility, of linear algebra in modern science and mathematics and some feel for way it is actually used in disparate subject areas. Twelve appendices are included. The last seven refer to MATLAB® code which, though not required and rarely mentioned in the text, can be used to augment understanding. For example, fifty-five MATLAB functions implement every tensor operation from Chapter 9. A zipped file of all code is available for download from the author's website.
Linear Algebra
Author: Meighan I. Dillon
language: en
Publisher: American Mathematical Society
Release Date: 2022-10-14
This textbook is directed towards students who are familiar with matrices and their use in solving systems of linear equations. The emphasis is on the algebra supporting the ideas that make linear algebra so important, both in theoretical and practical applications. The narrative is written to bring along students who may be new to the level of abstraction essential to a working understanding of linear algebra. The determinant is used throughout, placed in some historical perspective, and defined several different ways, including in the context of exterior algebras. The text details proof of the existence of a basis for an arbitrary vector space and addresses vector spaces over arbitrary fields. It develops LU-factorization, Jordan canonical form, and real and complex inner product spaces. It includes examples of inner product spaces of continuous complex functions on a real interval, as well as the background material that students may need in order to follow those discussions. Special classes of matrices make an entrance early in the text and subsequently appear throughout. The last chapter of the book introduces the classical groups.
Linear Algebra
Author: Sterling K. Berberian
language: en
Publisher: Courier Corporation
Release Date: 2014-07-16
Introductory treatment covers basic theory of vector spaces and linear maps — dimension, determinants, eigenvalues, and eigenvectors — plus more advanced topics such as the study of canonical forms for matrices. 1992 edition.