to the subject. Rank and nullity are defined, both in terms of linear transformations and in terms of matrices. The chapter then concludes with probably the most important topic of the book, isomorphism. Along with isomorphism, coordinates, coordinate maps, and change of basis matrices are presented. The section and chapter concludes with the discoveryofthe standard matrix of a linear transformation. Although there is more to come, a standing ovation for the standard matrix and its diagram would not be inappropriate.
Inner Product Spaces Although ℝn is usually viewed as Cartesian space, it is technically just a set of n × 1 matrices. Any geometry that it has was given to it in the second chapter, even though its geometry is a copy of the geometry of Cartesian space. A close examination reveals that the geometry of ℝn is based on the dot product. Mimicking this, an abstract vector space is given its geometry with an inner product, which is a function defined so that it has the same basic properties as the dot product. The vector space then becomes an inner product space so that distances, lengths, and angles can be found using objects like matrices, polynomials, and functions. Other topics related to the inner product include a generalization of the orthogonal projection, orthonormal bases, direct sums, and the Gram–Schmidt process.
Matrix Theory The book concludes with an introduction to the powerful concepts of eigenvalues and eigenvectors. Both the characteristic polynomial and the minimal polynomial are defined and used throughout the chapter. Generalized eigenvectors are presented and used to write ℝn as a direct sum of subspaces. The concept of similar matrices is given, and if a matrix does not have enough eigenvectors, it is proved that such matrices are similar to matrices with a nice form. This is where Schur’s Lemma makes its appearance. However, if a matrix does have enough eigenvectors, the matrix is similar to a very nice diagonal matrix. This is the last section of the book, which includes orthogonal diagonalization, simultaneous diagonalization, and a quick introduction to quadratic forms and how to use eigenvalues to find an equation for a conic section without a middle term.
As with any textbook, where the course is taught influences how the book is used. Many universities and colleges have an introduction to proof course. Because such courses serve as a prerequisite for any proof‐intensive mathematics course, the first chapter of this book can be passed over at these institutions and used only as a reference. If there is no such prerequisite, the first chapter serves as a detailed introduction to proof‐writing that is short enough not to infringe too much on the time spent on purely linear algebra topics. Wherever the book finds itself, the course outline can easily be adjusted with any excluded topics serving as bonus reading for the eager student.
Now for some technical comments. Theorems, definitions, and examples are numbered sequentially as a group in the now common chapter.section.number format. Although some proofs find their way into the text, most start with Proof, end with ■, and are indented. Examples, on the other hand, are simply indented. Some equations are numbered as (chapter.number) and are referred to simply using (chapter.number). Most if not all of the mathematical notation should be clear. Itwas decided to represent vectors as columns. This leads to some interesting type‐setting, but the clarity and consistency probably more than makes up for any formatting issues. Vectors are boldface, such as u and v, and scalars are not. Most sums are written like u1 + u2 + ⋯ + uk. There is a similar notation for products. However, there are times when summation and product notation must be used. Therefore, if u1, u2, …, uk are vectors,
and if r1, r2, …, rk are real numbers,
Each section ends with a list of exercises. Some are computations, some are verifications where the job is to make a computation that illustrates a theorem from the section, and some involve proving results where remembering one’s logic and set theory and how to prove sentences will go a long way.
Solution manuals, one for students and one for instructors, are available. See the book’s page at wiley.com.
Lastly, this book was typeset using LATEX from the free software distribution of TEX Live running in Arch Linux with the KDE Plasma desktop. Thediagramswere created using LibreOffice Draw.
Michael L. O’Leary
Glen Ellyn, Illinois
September, 2020
Acknowledgments
Thanks are due to Kalli Schultea, Senior Editor at Wiley, for her support of this project. Thanks are also due to Kimberly Monroe‐Hill and Gayathree Sekar, also at Wiley, for their work in producing this book.
I would like to thank some ofmy colleagues at College of DuPage who helped at various stages of this project. They are James Adduci, Christopher Bailey, Patrick Bradley, Jennifer‐Anne Hill, Rita Patel, and Matthew Wechter. I also thank my linear algebra classes of 2017, 2019, and 2020 who were good sports as I experimented on them with various drafts of this book. They found many errors and provided needed corrections.
On a personal note, I would like to express my gratitude to my parents for their continued caring and support; to my brother and his wife, who will make sure my niece learns her math; to my dissertation advisor, Paul Eklof, who taught me both set theory and algebra; to Robert Meyer, who introduced me to linear algebra; to David Elfman, who taught me about logic through programming on an Apple II; and tomy wife, Barb, whose love and patience supported me as I finished this book.
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/o’leary/linearalgebra
The website includes the solutions manual and will be live in the fall of 2021.
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