# A First Course in Linear Algebra

Ken Kuttler, Brigham Young University

Copyright Year: 2017

Publisher: Lyryx

Language: English

## Conditions of Use

Attribution

CC BY

## Reviews

This book covers a very large and comprehensive list of topics. Aside from the leading topics in a standard linear algebra course, there are some less-standard but highly important topics covered, such as spectral theory, abstract vector spaces,... read more

This book covers a very large and comprehensive list of topics. Aside from the leading topics in a standard linear algebra course, there are some less-standard but highly important topics covered, such as spectral theory, abstract vector spaces, curvilinear coordinates, and even a nice chapter on complex numbers (a topic which is often assumed even if students aren't so familiar with it). I wish there was a little more about other methods for solving linear systems, that is, anything other than Gaussian elimination/LU-factorization which should only be used as a last resort for large matrices, and is rarely used in practice. For example, a section on basic iterative methods (e.g., Jacobi or Gauss-Seidel) might be useful, if only to demonstrate to students that there are other, possibly better, methods out there. However, this is a very minor point, as most textbooks on linear algebra do not cover these topics either.

I did not find any major errors in the book.

Linear algebra itself will be a subject of high relevance for the far foreseeable future, and this book does a good job of capturing the major important points of what is now consider the classical core of linear algebra, and even extends a bit beyond this. However, modern linear algebra needs to be very mindful of cases in which matrices are extremely large (e.g., billion by billion or larger), as such matrices are becoming increasingly common in nearly all areas of science and engineering. Therefore, in sections where methods which are inefficient or unstable for large matrices, major warning signs need to be given. This text does not do a good job of this. For example, in the discussion of Cramer's rule, the only line that puts things into context is "Cramer’s Rule gives you another tool to consider when solving a system of linear equations." This is a major overstatement, as Cramer's rule has computational complexity O(n*n!), making it completely useless for solving anything larger than a 3x3 system. (Gaussian elimination, while still typically a bad choice, is at least only O(n^3)). Moreover, Cramer's rule is unstable even for 2x2 systems. Cramer's rule can occasionally be useful in theoretical applications, but the text does not discuss these nuances at all. Many other examples like this occur in the book, but this is perhaps the most glaring one.

The style of the book is nice and streamlined, cutting out much unnecessary fluff, while still clearly hitting the main points.

The textbook follows a nice, consistent build-up of definitions and notation. The colors and styles used (e.g., for the example boxes, the theorems, etc.) are also quite consistent.

The author did a really excellent job of separating the sections and making the as independent as possible. I also like that there seems to be modularity in terms of complexity. For example, a beginner can spend time reading Chapter 4, discussing linear algebra on R^n without getting bogged down in abstractions, while a more experienced reader can skip to Chapter 9 on abstract vector spaces, without feeling like they need to constantly return to Chapter 4. The exercises also seem independent. For instance, I did not find any places where an exercise relies on you having completed an exercise from an earlier section, which is very nice from the standpoint of modularity. I have seen other reviewers complain about the length of the text, but I think this may be a consequence of having a text that is so modular, and also inclusive of advanced topics. I would much rather have a long text from which I can extract the pieces I need, than a short text with topics all mashed together. I think the author was wise to trade brevity for modularity in this book.

This text has a clean, clear development of linear algebra. Another reviewer said that the text introduces subspaces before Vector spaces, but this does not really seem quite the whole story: Subspaces of R^n are introduced, *then* much later, abstract vector spaces are introduced. This seems to me to be a very reasonable way to build things up. Even historically, people first started to understand subspaces in R^n before they understood the idea of an abstract vector space. I think it really is a nicely organized text, with easy-to-find topics, a carefully layered build-up of topics, and with the main topics cleanly separated.

I had no problems navigating the pdf. Both text and graphics appeared very clear, rendered immediately, and were easy to read.

I found no grammatical errors.

This is a book on abstract mathematics, and as expected, no material arises that would reasonably be considered to be ulturally insensitive or offensive.

Very nice book, and I am considering using it in a classroom, but I wish it had a little more of a nod to the difficulties of handling large matrices. For a classically-focused course though, the book is excellent.

For the book's stated purpose of providing a first approach to linear algebra is met. The rigor is appropriate and the author has gone to great lengths to cover the standard definitions, theorems, and examples that are at the heart of linear... read more

For the book's stated purpose of providing a first approach to linear algebra is met. The rigor is appropriate and the author has gone to great lengths to cover the standard definitions, theorems, and examples that are at the heart of linear algebra.

I have found no errors during the first time I taught from this book.

The author has done an exceptional job providing a modern exposure to linear algebra that is accessible to students of all STEM disciplines. The applications are appropriate for the audience and the theory of linear algebra will not become obsolete in the next twenty years.

Consider the following passage from an example on row reduction: "Notice that the first column is nonzero, so this is our first pivot column. The first entry in the first row, 2, is the first leading entry and it is in the first pivot position." The text is clear, accurate and not verbose. This style is consistent throughout the text and technical terminology is properly used. I am also happy to say that the author sets a fine example of never abusing notation.

This book provides a consistent and mature approach to the topics in each section. In particular, the final section on vector spaces is written at a level that is appropriate for students that want to take a deeper look at the underlying frameworks of linear algebra.

The challenge in a linear algebra text is that there are so many definitions to cover in order for the abstract theory to develop. The first few section of the book, 1-4, are essential. However every topic in sections 1-4 need not be assigned. For example, section 2.2's discussion on LU factorization is not necessary and can be omitted without any disruption. Once the first 4 chapters are covered it is trivial to reorder the later sections to fit the objectives of your own course. Students at this level should have no problem filling in the small gaps caused by any reordering.

The book takes a standard approach for covering systems of equations, developing matrix theory, determinants, properties of R^n, linear transformations, the eigen problem, and ends with a deeper dive into general vector space theory. Each section provides simple, leading examples that explore the new topic. Theorems, and when necessary, their proofs are presented in a logical fashion. There are a sufficient number of examples for each section covering the basic ideas and cases that one encounters in linear algebra. From a personal opinion, section 4, R^n, is dense and could easily be broken up into two parts. The first on the geometry of R^n and the second on the notion of linear independence and orthogonality.

Theorems, examples and figures are clearly denoted throughout the textbook. The use of color is also consistent and makes it easy to skim the sections. I appreciate that learning outcomes are well organized and appropriate for each section. I would suggest adding a link in the table of contents to the exercise section of each chapter.

This book's style and grammar is consistent with the books found from the major textbook publishers.

This book does not cover any topics about culture, race or ethnicities and is written in a respectful manner.

I have taught from the author's related book, "Elementary Linear Algebra", and I find the refinements here of the material to be a positive change in every regard. This book is an excellent example of a mature textbook for students in STEM fields.

The book is sufficiently comprehensive for its purpose as a first course in the subject. The row reduced echelon form and its many consequences and applications are covered well in the first several chapters. Later chapters on linear... read more

The book is sufficiently comprehensive for its purpose as a first course in the subject. The row reduced echelon form and its many consequences and applications are covered well in the first several chapters. Later chapters on linear transformations and spectral theory are presented at a nearly ideal level for such a course and a reasonable selection of applications are included.

Aside from a few minor typographical errors the content appears to be very accurate. No substantive errors were noted.

The notation and presentation is similar to other recent books on linear algebra.

The book is written in a very conversational style, and for the most part this aids the presentation. Dense paragraphs are mostly avoided and the text is broken up with examples, theorems, etc... in a similar manner to other modern textbooks.

I found no inconsistencies aside from a few minor issues with incorrect references to theorem numbers (ex. theorem 9.35 is referenced in the proof of theorem 4.83 when the intended reference is clearly from chapter 4).

The length of individual sections is reasonable and topics are as self-contained as they can be given the nature of the subject. For example, a student who already knows about vectors in R^n from vector calculus or another course could start chapter 4 at section 10 with only minor difficulties.

The order and presentation of topics are quite clear, with plenty of examples. As noted by another reviewer, the overall length of the text may seem excessive to some readers.

The small number of figures in the text serve their purposes well. Overall, the text is very easy to navigate and visually attractive.

No significant grammatical errors were noted.

No issues here.

This book contains all of the material that would generally be covered in a Freshman or Sophomore Linear Algebra course. The section on vectors is quite extensive, and would be excellent to use in a Freshman course that needed to introduce vectors... read more

This book contains all of the material that would generally be covered in a Freshman or Sophomore Linear Algebra course. The section on vectors is quite extensive, and would be excellent to use in a Freshman course that needed to introduce vectors very early for use in Engineering courses. On the other hand, the sections on Linear Transformations and Eigenvalues are exactly what I would want to see in a Sophomore course that was designed more for mathematicians. The section on abstract vector spaces I find somewhat deficient for a more theoretical course.

The content is fully accurate. The theorems and proofs that are provided in each section are presented with precision, and yet easy to read.

The fundamental courses of mathematics are not generally given to change, but Linear Algebra perhaps more than the rest does need to keep itself updated with the rapidly changing field of Numerical Analysis. I think that Kuttler has done an excellent job of keeping up with the current methods, but has not written in such a way as to make the text dependent on any particular numerical methods that are in vogue.

I like very much the style of writing and even the formatting of headers and content titles. I found it very readable and easy to find what I was looking for.

I did not detect any inconsistency in the way that the material was presented. The only thing that might be considered an inconsistency is that Subspaces and presented in the text prior to vector spaces. It is somewhat unnatural to define the subspace of something that is not defined as yet, but sadly it has become somewhat commonplace to do things in this way.

The sections of the book are easily adaptable. I did like the fact that the section on vector spaces was written in a way to be included earlier if desired.

Other than this issue with the section on vector spaces, I found the organization and flow of topics to be quite natural. Each topic comes in its proper place, but not in such a way as to detract from its adaptability.

I like the way that sections and headings and theorems all are very descriptive, but also numbered, so that they can be easily found.

I did not find any grammatical errors.

I don't think there was any instance of cultural insensitivity.

Exercises are provided at the end of each major section, and I found them to be ample both in quantity and in terms of the level of difficulty.

In my experience, text book works extremely well with the learning outcomes defined by my institution for entry level linear algebra course. For my students, textbook provides a foundation for the course. Techniques to solve the problems are easy... read more

In my experience, text book works extremely well with the learning outcomes defined by my institution for entry level linear algebra course. For my students, textbook provides a foundation for the course. Techniques to solve the problems are easy to follow and build upon as the topics gets harder

Upon an initial review I found no obvious errors.

Content is up-to-date and should stay relevant for a long while

Text provides detailed instruction to solve the problems.

Text stays consistent throughout with definitions, solutions and responses. Students get used to the pattern of solving the problems

Text book can be taught using sections and subsections without creating much confusion. Few chapters that are interconnected may need extra care to rearrange since students would need to have some basic understand of the concept.

Flow of the chapters fits the learning outcomes needed to incorporate.

I have viewed it online only and it works very well on a computer screen.

Upon initial review, No spelling or grammar errors were encountered.

Math has a great flexibility when it comes to being culturally relevant. An inclusion of socially conscious everyday problems may help students with the following question- when would I use this math in real life.

N/A

The book includes all the topics we require in our introductory linear algebra course. read more

The book includes all the topics we require in our introductory linear algebra course.

I couldn't find any errors in accuracy.

The content is up-to-date and includes applications that are relevant to many of the students' future plans.

Very well written, clear explanations & lots of examples.

I found no internal inconsistencies except for the notation used for "solution set" on p. 17.

The sections of the book are of a reasonable length and the organization makes sense.

The topics flow very nicely.

Easy to navigate & especially useful to have internal links. Images were nicely done.

I found no grammatical errors, but a typo on p. 70: Above Example 2.20, it reads ...product AB maybe be...

Not offensive, but could have included examples/exercises that were multicultural.

I plan to propose that we adopt this text as our required text for our introductory linear algebra course. On p. 293, when defining basic eigenvectors, I would like to see them referred to as "basic eigenvectors associated with lambda". Then the following sentence is true. I find the title of Cor. 9.28 confusing; maybe "length of bases" would work better.

This text covers all the material an instructor could want to include in an introductory Linear Algebra course. The first three chapters (Systems of Equations, Matrices, and Determinants) are standard in any introductory Linear Algebra course,... read more

This text covers all the material an instructor could want to include in an introductory Linear Algebra course. The first three chapters (Systems of Equations, Matrices, and Determinants) are standard in any introductory Linear Algebra course, but the content of the remainder of such courses varies quite a bit. The subsequent chapters of this book are each pretty well self-contained, so it would be pretty easy to adapt the content to a particular curriculum. There is no glossary, and the index, while short, seems to be comprehensive.

The book is over 400 pages, so I have not proof-read the entire book. However, a few hours of reading revealed no errors or inaccuracies. It is clear the author took great care with the presentation of the material, so I didn't expect there to be a significant number of errors.

The material is so fundamental in mathematics, and this book covers all the important topics. Relevance/longevity will not be an issue.

The clarity of the writing is what I find most appealing about this book. The proofs are all included and easy to read. This book would be suitable for students' first exposure to proofs. There are also plenty of thorough examples. Terminology is always an issue with students in this subject, but the author has used a color scheme to identify definition boxes in the text and differentiate them from examples, theorems, etc. Linear Algebra texts often suffer from aggressive detail paid to procedure and computation. This book includes a lot of prose to motivate technical procedures. While it increases the length of the text, it is done very well.

For this particular subject, consistency in terminology is essential. There is some redundancy for the purposes of increasing modularity of the latter half of the book, but the consistency of the terminology and framework of the book is nonetheless first-rate.

The modularity of this book is quite good, and this is of particular importance for this particular subject. One could very easily reorder the chapters of the book to fit their curriculum.

The presentation of the topics in this book is thorough almost to a fault. While the exposition is quite clear and there are many great examples and explanations, the overall length could be intimidating to some students. To cover all the material one in an average one semester introductory Linear Algebra course, one could have over 300 pages of mathematics text for their students to read. Depending on the course/students, this could be an issue. Despite its length, though, it is both extremely well-organized and easy to read.

The diagrams in this book are great, though there aren't a lot of them in the first half of the book. The geometric intuition in this subject is extremely important and difficult to convey, and this book does a sufficient job. This is in part due to the great modularity of the book. The geometric interpretations of matrices and determinants are left for the chapters in the second half of the book; it makes for a very algebra-heavy first three chapters. Navigation in the book is very good. I would like to have all terminology hyperlinked to its definition box, but other than that, everything else is hyperlinked.

I found no grammatical errors.

Cultural relevance is not an issue for this book.

I like this book quite a lot. For instructors dissatisfied with their ability to reorder standard Linear Algebra texts to suit their needs, this provides a very nice alternative. It would be easy to adapt to any introductory curriculum. There are plenty of very nice exercises. The one missing element that I like in Linear Algebra exercises is True/False; aside from supplementing for that, one could use these exercises exclusively. There is no use of technology (MATLAB, Maple, calculator, etc.) integrated into this text.

## Table of Contents

1 Systems of Equations

- 1.1 Systems of Equations, Geometry
- 1.2 Systems Of Equations, Algebraic Procedures

2 Matrices

- 2.1 Matrix Arithmetic
- 2.2 LU Factorization

3 Determinants

- 3.1 Basic Techniques and Properties
- 3.2 Applications of the Determinant

4 R^n

- 4.1 Vectors in R^n
- 4.2 Algebra in R^n
- 4.3 Geometric Meaning of Vector Addition
- 4.4 Length of a Vector
- 4.5 Geometric Meaning of Scalar Multiplication
- 4.6 Parametric Lines
- 4.7 The Dot Product
- 4.8 Planes in R^n
- 4.9 The Cross Product
- 4.10 Spanning, Linear Independence and Basis in R^n
- 4.11 Orthogonality and the Gram Schmidt Process
- 4.12 Applications

5 Linear Transformations

- 5.1 Linear Transformations
- 5.2 The Matrix of a Linear Transformation I
- 5.3 Properties of Linear Transformations
- 5.4 Special Linear Transformations in R^2
- 5.5 One to One and Onto Transformations
- 5.6 Isomorphisms
- 5.7 The Kernel And Image Of A Linear Map
- 5.8 The Matrix of a Linear Transformation II
- 5.9 The General Solution of a Linear System

6 Complex Numbers

- 6.1 Complex Numbers
- 6.2 Polar Form
- 6.3 Roots of Complex Numbers
- 6.4 The Quadratic Formula

7 Spectral Theory

- 7.1 Eigenvalues and Eigenvectors of a Matrix
- 7.2 Diagonalization
- 7.3 Applications of Spectral Theory
- 7.4 Orthogonality

8 Some Curvilinear Coordinate Systems

- 8.1 Polar Coordinates and Polar Graphs
- 8.2 Spherical and Cylindrical Coordinates

9 Vector Spaces

- 9.1 Algebraic Considerations
- 9.2 Spanning Sets
- 9.3 Linear Independence
- 9.4 Subspaces and Basis
- 9.5 Sums and Intersections
- 9.6 Linear Transformations
- 9.7 Isomorphisms
- 9.8 The Kernel And Image Of A Linear Map
- 9.9 The Matrix of a Linear Transformation

A Some Prerequisite Topics

- A.1 Sets and Set Notation
- A.2 Well Ordering and Induction

B Selected Exercise Answers

## About the Book

This text, originally by K. Kuttler, has been redesigned by the Lyryx editorial team as a first course in linear algebra for science and engineering students who have an understanding of basic algebra.

All major topics of linear algebra are available in detail, as well as proofs of important theorems. In addition, connections to topics covered in advanced courses are introduced. The text is designed in a modular fashion to maximize flexibility and facilitate adaptation to a given course outline and student profile.

Each chapter begins with a list of student learning outcomes, and examples and diagrams are given throughout the text to reinforce ideas and provide guidance on how to approach various problems. Suggested exercises are included at the end of each section, with selected answers at the end of the text.

Lyryx develops and supports open texts, with editorial services to adapt the text for each particular course. In addition, Lyryx provides content-specific formative online assessment, a wide variety of supplements, and in-house support available 7 days/week for both students and instructors.

## About the Contributors

### Author

**Ken Kuttler, **Professor of Mathematics at Bringham Young University. University of Texas at Austin, Ph.D. in Mathematics.