Linear Algebra in Context: A Gateway to the Discrete Fourier Transform, Bilinear and Sesquilinear Forms, Algebras, Tensors and Mechanics (Springer Undergraduate Texts in Mathematics and Technology) - Hardcover

Susanka, Lawrence

Hardcover

ISBN 10: 3031784235 ISBN 13: 9783031784231

Publisher: Springer, 2025

Synopsis

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.

About this title

About the Author

Lawrence Susanka received his PhD in Mathematics in 1991 from the University of Minnesota. He was tenured at Bellevue College for 28 years until 2019. He currently helps manage the Math and Physics Colloquium series at Bellevue. Dr. Susanka has an active interest in physics and mathematics, rock and ice climbing, chess problems, horses and running mountain trails.  He is a supporter of The Planetary Society, various skeptical organizations such as the Committee for Skeptical Inquiry, and is a fan of science fiction in its myriad forms.

From the Back Cover

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.

Edition details

• Publisher: Springer
• Publication date: 2025
• Language: English
• ISBN 10: 3031784235
• ISBN 13: 9783031784231
• Binding: Hardcover
• Number of pages: 732