# Linear Transformations on Vector Spaces

Scott Kaschner, Butler University

Amber Russell, Butler University

Copyright Year:

Publisher: PALNI

Language: English

## Formats Available

## Conditions of Use

Attribution

CC BY

## Reviews

The book presents a very theoretical view of linear algebra. There are a lot of theorems, corollaries, and even some algorithms. What is missing from the text are applications. The authors make this comment on pg iii, "You are not learning an... read more

The book presents a very theoretical view of linear algebra. There are a lot of theorems, corollaries, and even some algorithms. What is missing from the text are applications. The authors make this comment on pg iii, "You are not learning an applied version of Linear Algebra. The goal of this text is to give you a strong foundation in this topic so that you can recognize the applications in your own field as you encounter them." In my experience, young undergraduate students do not recognize when they can apply linear algebra without seeing examples first.

There are a few sections that are application based, but even these are lacking. Section 4.8, a section graphics applications has a single graphic, which is a fractal and not an applied example. The example could have easily been something related to optical character recognition. In Chapter 5, the chapter on eigenvalues and eigenvectors, the text says, "Let’s discuss an application of all this." (pg 355). The application is to diagonalize matrices. There is no discussion about how diagonalization might be applied. I would not classify this as an application in the traditional sense. It is more theory.

If your students are motivated by application, the crowd that asks, "When are we going to use this?" they will not be motivated by this text. This is unfortunate because, as the authors state on pg iii, "Linear Algebra is inarguably one of the most applicable areas in modern mathematics."

I found no errors. There are examples that show steps. The homework questions reflect the examples. A few thoughts are incomplete and left for the reader to finish. While this is typical in mathematics texts, this text does this more than I am accustomed to.

The concepts for a beginning linear algebra course have remained relatively consistent for decades. This text has the important topics. The lack of applications makes it unlikely that it will go out of date, though the informal language will feel dated in a few years. Or, to quote one of my students, "why are boomers trying to sound cool?" It does seem, though, that any updates will be easy to implement.

The book is written with an informal voice (or a silly voice as the authors state on pg ii). For example, on pg 331, "Yay! We found an approximate eigenvalue and eigenvector." There are also comments in the margins, intended to keep the interest of students. However, students who I had look at the book found the comments in the margins more distracting than useful. Students felt like they had to read the comments in the margins, as if they were essential to the text. Sometimes the notes in the margin are useful (like on page 30 where the margin comments are essential to understanding the text) but other times you have exchanges like this on pg 328: "I shall smite thee with my Matrix Powers! No! Not the Matrix Powers! Ahh!" Students struggled to understand why these comments were there. The students in my class did not find them entertaining. My students found the unicorns obnoxious. Perhaps they make more sense with the way the authors present the material in their classes. However, if you are not the authors, you will have to address the unicorn in the room.

The notation and formatting of the text is very consistent. Terminology is both standard and consistent throughout. The use of colors is helpful and consistent. The margins are always on the right side, which is problematic when printing double-sided and putting in a binder.

In the comment to all readers it says, "We encourage you to follow the sections in order." The authors offer an alternative organization of the material in the note to the reader, however that sequencing references sections that have not been covered yet, in their alternative sequencing. For example, the alternate sequencing suggests starting with section 3.4. However, the first theorem in section 3.4 relies on a definition in section 3.2, which the authors suggest skipping until later using the alternate organization. You do need to follow the text in the order presented and

If my goal was to teach students vector spaces, this book would be organized perfectly. If the goal is to teach applications of linear algebra to students majoring in engineering, computer science, physics, and chemistry, this book will cause confusion with the organization. Knowing the audience of the book is essential.

In the PDF version, there are large blank spaces. I presume these are so students can write notes and do the exploration activities. If students are using the PDF version on a device, I'm not sure how useful the spaces are. In some cases the blank space is either too big or too small for what I would like to put in the space. If students print, that creates another problem.

There are incredibly wide margins that occasionally have notes always on the right hand side. However, a lot of pages do not contain notes in the margin, just leaving 2-3 inches of blank space on the side of each page. There are instances where the main text flows over to the margin (pg 44, caption to figure 1.9 as an example). There are pages that have a single line of text on them, otherwise blank (pg 161 as an example). This means the overall document likely takes many more pages than necessary, contributing to increased printing costs for students who have accommodations that require a printed text.

In the glossary, when a definition ends with a formula, the period and page reference appear on the next line alone (pg 416, definition of basis is the first example). Students sometimes had difficulty recognizing which word the link was for (the one above or below). There are also two periods at the end of each definition in the glossary and sometimes there is a space between the periods (coefficient matrix as an example) and other times there is not a space (codomain as an example).

As mentioned previously, my students found the unicorns distracting.

I found very few grammatical errors (pg 48, marginal note 20 says "RIcky" with a capital I for some reason) As mentioned previously, the tone is very informal. The text has a lot of ellipses ... and incomplete thoughts ... encouraging the reader to finish the thought. This is not bad, but students need to be prepared to read a text that does not finish thoughts.

The text is completely devoid of any mention of race, ethnicity, or backgrounds. There is no historical information presented. It is not offensive; it avoids the topic. The text references Hermitian matrices without mentioning Charles Hermite. It mentions Jordan block and Jordan chain without mentioning Camille Jordan. There is a great opportunity to use the marginal notes to mention why Hermitian, Jordan, Markov, and Gauss are capitalized.

Overall, this review comes across as more critical than my intention. If you can find the right audience for this book, I think it is a good book.

I struggled to find the right audience. Third semester college students majoring in engineering, chemistry, or computer science, or physics struggled to find motivation with this text. There are no applications to relate it back to their disciplines, and the authors admit this early on. I spent a lot of time answering questions of "But when am I going to use vector spaces in chemistry?" and "What's the point of a basis in computer programming?" As the instructor, I enjoyed and appreciated the book. My students, though, bought or borrowed copies of the text we previously used to teach the class so they could, "understand what you are teaching."

I think the content is good and the organization makes sense if your goal is to teach the theory of vector spaces. If you want applications, though, this book is not the book you want. If you have students who are planning further study in mathematics, this is likely the right book to use.

## Table of Contents

- Preface
- Chapter 0: Functions on Sets
- Chapter 1: Vector Spaces
- Chapter 2: Bases
- Chapter 3: Linear Transformation
- Chapter 4: More Fun with Matrices
- Chapter 5: Square Matrices and Invariant Subspaces
- Appendix
- Glossary

## Ancillary Material

Submit ancillary resource## About the Book

Linear Transformations on Vector Spaces serves primarily as a textbook for undergraduate Linear Algebra courses. While standard Linear Algebra books begin by focusing on solving systems of linear equations and associated procedural skills, our book begins by developing a conceptual framework for the topic using the central objects, vector spaces and linear transformations. It covers the same concepts, skills, and, applications as conventional texts in a one-semester course, but students walk away with a much richer and more useful mastery of the topics. The book is structured to facilitate the implementation of the flipped classroom. The text features a continuous narrative to illuminate the big picture of the material and is written to help students develop their textbook reading skills. Also, there are “Explorations” scattered throughout each section; these are quick first examples intended for students to complete while reading before class meetings. Additional materials include section overview homework assignments and worksheets that can be used for in-class practice.

## About the Contributors

### Authors

**Scott Kaschner** is an Associate Professor and the Department Chair in the Department of Mathematical Sciences at Butler University. He earned a Ph.D. in Mathematics from Purdue University (at Indiana University Purdue University Indianapolis) and did a postdoc at the University of Arizona. His primary research area is complex dynamical systems in one or several variables; he also does work in operator theory, undergraduate mathematics education, professional development for mathematics educators, and mathematical biology.

**Amber Russell** is an Associate Professor in the Mathematical Sciences Department at Butler University and the current Director of Butler’s Engineering Dual Degree Program. She received her Ph.D. in Mathematics from Louisiana State University. Her research area is geometric representation theory, but she particularly enjoys studying the combinatorics associated to Springer theory. She was a postdoctoral researcher for the Algebra Research Group at the University of Georgia, where she first taught Linear Algebra, both an applied and a theoretical version. She is responsible for all unicorn images in this text, including the really unfortunate original one in Chapter 2.