A First Course in Linear Algebra
Robert Beezer, University of Puget Sound
Pub Date: 2008
Publisher: University of Puget Sound
Conditions of Use
This book contains a standard set of topics one would expect to see in a first semester Linear Algebra course, beginning with systems of linear read more
This book contains a standard set of topics one would expect to see in a first semester Linear Algebra course, beginning with systems of linear equations and transitioning into vectors and matrices. Abstract vector spaces appear in the middle of the book once students are well-equipped to make the transition from real or complex vector spaces. The appendix provides a good review of complex numbers and basic set theory. The book essentially ends with orthonormal diagonalization. I rank among those that would consider quadratic forms and singular value decomposition as unfortunate omissions from the text. As other reviewers have pointed out, the acronym-labeling style for theorems is odd. Proofs and examples are usually done in sufficient detail, but the labeling system makes it more difficult than necessary to find references to other theorems. There is a list in the appendix of where to find theorems but using this seems like an unnecessary step for students. In addition, the index of definitions in the appendix is sorted by section, rather than alphabetically, making finding definitions cumbersome.
I found no major errors in the text.
This book can certainly compete with other standard Linear Algebra texts, such as Lay’s “Linear Algebra and its Applications”. The Sage supplement makes it especially relevant for instructors who wish to implement computation/programming into their course.
The book seems relatively easy to read for students. Proofs are generally given with great detail and references that students can use to understand individual steps. Sometimes, the author includes elements in a proof that belong more in the discussion before or after. For example, the proof that the inverse of a matrix product is the product of the inverses in the opposite order begins with an analogy to dating services. It is a cute analogy, but does not give students a good example for how formal proofs should be written.
The author does a good job of maintaining style throughout.
While there is some natural dependence between topics, I saw no reason that many of the chapters could be moved around at will. Of course, it would make no sense to move the discussion of determinants after that of eigenvalues/eigenvectors. The author introduces the idea of a basis in the chapter on vectors and even uses the term without fully defining it. This is saved until the chapter on abstract vector spaces but does prevent that chapter from being moved until later in the book, presenting some modularity problems for those that choose to focus primarily on R^n.
The structure of this book is what I would expect in an introductory Linear Algebra text.
There is a complete lack of figures/diagrams in this text. Not only does this obscure geometric concepts related to Linear Algebra, it also makes the text less inviting for students.
I found no major grammatical issues in the text.
This text does not heavily emphasize applications and therefore it is hard to judge the cultural relevance of the text. In fact, Linear Algebra is such an important part of modern mathematics/computer science/engineering that the text does some students a disservice by not focusing on these.
Because this book is open source, I would consider using this in a future Linear Algebra course, possibly supplemented with my own notes or other resources.
If anything, this textbook is too comprehensive: it exhaustively covers all linear algebra canon. read more
If anything, this textbook is too comprehensive: it exhaustively covers all linear algebra canon.
I found no errors after a month of teaching out of the book.
This book attempts to balance its relevance among three audiences: general engineers, mathematics majors, and computer science students. At times it seems to swing too far in the direction of math major, but it remains a valuable resource for all audiences.
Many of the examples were large systems meant to be presented on a computer rather than a blackboard. This emphasizes the material's applicability but at the expense of efficiency.
It is consistent.
I taught only portions of this book as part of a multivariable calculus course, and the book's modularity held up very well.
The topics were clear for me as an instructor who already knew the material, but I don't know if my students were able to follow the text. However, reading mathematics texts is an acquired still and there are very few fully accessible to undergraduates.
The only feature I was missing were bookmarks in the pdf of the solutions manual, but that was a big one.
I found no errors.
Is math a culture?
As a resource for a multivariable calculus class, this book fit my needs perfectly. I would consider using it for a full-semester linear algebra course after my experience.
This is a great book that covers most topics that should be included in an introductory linear algebra course. In fact, many of the topics are read more
This is a great book that covers most topics that should be included in an introductory linear algebra course. In fact, many of the topics are discussed in more depth than what is necessary for an intro course. The Reading Questions at the end of each section make this book easy to use for a flipped style course. The sections on complex number operations, set theory are nice additions that help students gain a better understanding of these topics. The section on proof writing techniques is especially useful for students who have not had much exposure to proof writing. However, some topics that I usually cover in intro linear algebra like LU decomposition and applications to computer graphics are not included in the book.
The book is well-written and very accurate.
The content is very relevant and up-to-date. However, I think that more applications should be addressed, especially relating to the use of linear algebra in image processing and computer graphics.
The book is very clearly written and the style is easy to read.
The text is internally consistent in terms of terminology and framework.
The text is easily and readily divisible into smaller reading sections that can be assigned at different points within the course. Each section can be covered in an hour long class if the students do the reading and complete the Reading Questions in advance.
The book is well organized and the topics are presented in a logical manner. I wish that the acronyms used were more suggestive to make it easier to remember what they stand for. I like that the exercises have a letter indicating their type (e.g. T for theoretical), but don’t quite understand how they are numbered. For example, one section has exercise T12 immediately followed by exercise T20.
I find the online version the easiest to navigate followed by the pdf version. I like that the electronic and online versions have hyperlinks that make it easy to find references. However, the use of acronyms for the names of chapters, sections, examples, definitions, and theorems makes navigation more challenging. For example, it is difficult to go back to where you were after clicking on a hyperlink in the pdf because the sections are not numbered, so one can easily get lost. The frequent referencing using acronyms only (instead of pages numbers) makes it very difficult to use the print version.
I have not found any grammatical errors.
I have not found anything that is culturally insensitive or offensive in the text.
This is a high quality open source textbook that I would strongly recommend to any instructor teaching an introductory linear algebra course. The website corresponding to the book has plenty of supplementary resources for both students and instructors. The solution manual includes detailed answers for almost all the exercises. As an instructor, I wish that there were more exercises for which the students could not download the answers for to make assigning homework easier.
I examined this book carefully last semester while searching for a good inexpensive (or free) textbook to adopt for a sophomore-level linear algebra read more
I examined this book carefully last semester while searching for a good inexpensive (or free) textbook to adopt for a sophomore-level linear algebra course. This book contains all the topics that I'd normally cover in such a course, plus more. The prose is often conversational, but ultimately accurate, unambiguous and lucid. The book does have some quirks, the most noticeable of which is an extensive reliance on acronyms. Chapters are not numbered, but rather tagged with sometimes cryptic abbreviations. For example, the book begins with Chapter SLE (Systems of Linear Equations), followed by Chapter V (Vectors), then Chapter M (Matrices), etc. Even theorems, definitions, examples and diagrams are designated in this way. For instance, Definition ROLT is for the rank of a linear transformation. (Why not just call it "Definition RANK"?) All this takes some getting used to, but such brevity may have a place in classroom exposition. If there is a serious omission, it is that the book has scarcely any figures at all, which is surprising given the geometric nature of linear algebra. And I felt that the occasional figures fell short of really illuminating the ideas that they were supposed to convey. For example, Diagram NILT (non-injective linear transformation) is identical to Diagram NSLT (non-surjective linear transformation), except for labeling. So by themselves they don't clearly differentiate the two ideas. Further, these illustrations show generic point sets, not vector spaces. I'd be more comfortable seeing (say) non-surjectivity illustrated by a map from 2-D space to a plane in 3-D space, etc. I had trouble locating an index in the on-line version of the book.
I found no mistakes at all.
I believe the book is very up to date. Some instructors may want to see a little more on matrix decompositions, but this is not an issue with me. Regardless, because of the non-numeric labeling of chapters and definitions, it would be very easy for the author to add material without affecting the numbering of subsequent sections. For this reason, I rank the book's longevity as high.
The prose is very clear, and one feels that it has been informed by many years of teaching the subject. As mentioned above, I believe that it would be even clearer with the addition of well-crafted figures.
The author has done an excellent job here. The book is remarkably uniform in tone and format, and is uniquely Beezer's work from beginning to end. He has created his own brand of textbook.
The book is broken into sections and subsections, and theorems, proofs, definitions and examples are clearly delineated. The acronym labeling scheme makes the book feel especially modular, possibly at the expense of emphasizing the interdependency among the various topics.
The sequencing is perfectly logical and natural, and l would see no reason to do anything in a different order. This is one instance where the acronyms seem out of place, as a simple numeric labeling of the chapters would underscore the importance of the flow of ideas in a way that the acronyms do not.
I read the online version, which I thought was pretty good. I did find some aspects of the experience to be slightly disconcerting. For example, it's hard to gauge how long a section will be when clicking on an example can suddenly expand a simple phrase to an entire page, or more. But whatever problems I had may have been due to my own preference for thumbing through paper books.
I found no problems with the grammar.
It is difficult to imagine how linear algebra could be culturally insensitive. At any rate, I can't imagine that the author has offended anyone.
There is a lot of great basic material here. However, there are several topics missing that I would consider part of a standard first course in read more
There is a lot of great basic material here. However, there are several topics missing that I would consider part of a standard first course in linear algebra. Matrix factorizations, such as the Cholesky factorization, or decompositions, such as the LUD decomposition, do not appear to be treated. The singular value decomposition has achieved an important status in linear algebra and it should be found even in first courses. Strang's Linear Algebra did not have principal component analysis in 1984, but it does now for example. There is no index in this book that I can find.
The book appears very carefully written and accurate.
The basic material will not change and as such this text could be used a 100 years from now. However, it is missing more applied ideas, such as linear algebra in image processing, that are becoming increasingly popular and serve to decrease its relevance.
The writing style is very clear. However, the use of abbreviations such as Theorem SLEMM, and Definition NM, make the book harder to read. They also not very suggestive as a mnemonic device.
Yes, the book appears very consistent.
The topics are very nicely modular, but I would probably rearrange the order in which they are taught.
The topics are presented in a carefully thought out manner and the structure is reasonable. I would probably want matrix multiplication defined before introducing the solution of linear systems.
The hypertext links are great. Navigating the text is a pleasure. The inclusion of Sage is also a huge addition.
The grammar is fine.
The text is culturally relevant and not offensive. Unless you happen to not like matrices ;-)
Nice addition to the available resources that I am sure will be attractive for a lot of instructors.
The text covers all the topics of a first course in linear algebra. There is discussion on set theory, complex numbers and proof techniques. Complex read more
The text covers all the topics of a first course in linear algebra. There is discussion on set theory, complex numbers and proof techniques. Complex number are mentioned very early in the text although not used. Very little emphasis on the geometric approach and more leaning to operations research.
I found an error early on in the reading under Proof Techniques D "A definition is usually written as some form of an implication, such as “If something-nice-happens, then blatzo.” However, this also means that “If blatzo, then something-nice-happens,” even though this may not be formally stated." I did not come across any other errors, although I didn't edit the entire book. Sage is used and there is a section that calls out a video that was not accessible. There were other issues with the Sage tutorial, the "blue line" did not appear for instance.
The book is written using Sage which we do not use t this time. The Sage Cell Server is nice and allows students to use Sage without downloading it. The book is such that Sage is not required. Since the book is editable and Sage is also an open resource I see no problem with the longevity of this OER. The examples used in the text are relevant and up to date.
The writing in the book is very clear. Many examples help put the mathematics in context in each section.
The book is very consistent in terminology and structure. Each section has subsections with a description, example(s), reading questions and exercises. The reading questions are designed to be completed by the student before class on the topic with most of the exercises having worked out solutions.
The text is modular and could be reorganized but it flows by topic in such a way as not to be necessary. The proofs and their descriptions could be left out for a very early course in matrices. The online version has so many hyperlinks that it became a bit confusing where I had left off and how to get back.
The flow and structure was ok as long as I didn't click on too many hyperlinks. I found the hyperlinks lead to good examples and definitions but with no chapter/section numbers it was difficult to go back. Seemed like a lot of jumping around leading me to get a bit lost and having to reopen sections I'd already read. Because of the acronym section names O could come after V. I found that experience a bit frustrating and decided bypass this feature.
The displays and charts came across just fine. As far as navigation, it may be the operator but see my answer to number 7. Some of the links opened up a window that allowed the user to continue reading. This was true for most examples but not when directed to another section.
The grammar was fine, easy to read.
I did not fine anything insensitive or offensive in this book.
I liked the book overall. I like the printable flash cards for students in the supplemental section. The use of archetypes is also very useful and aides in understanding. I did not do well with all the acronyms, SLE vs SSLE vs SSSLE, section CNO with subsections CNE or CNA. Too many of these for me, I would suggest numbers. Like Section 2: Vectors, Section 2.1: Vector Operations.
This book includes a good selection of topics for a semester-long linear algebra course. read more
This book includes a good selection of topics for a semester-long linear algebra course.
I did not notice any errors. The open-source model allows any errors to be corrected promptly.
Most of the material is basically timeless. The book does include computer code that can be used with SageMath, an open-source computer algebra system. Because SageMath is open-source, it should be possible to obtain a copy indefinitely.
The book is easy to read, with practical examples sprinkled throughout. In addition, in the electronic version, the interface makes it easy to refer to previous theorems or examples.
The author uses consistent terminology and notation throughout.
The book is more modular than most other math texts. For example, theorems are not numbered, but given abbreviations, so that they would not need to be renumbered should you choose to adopt and incorporate sections into another text. Of course, some sections depend on results or material from others, which cannot be avoided in a math text. (But even then, the interface makes referring to the previous material easy.)
The book is organized well. The author moves from concrete to more abstract concepts, starting with matrices and column vectors before moving on to abstract vector spaces and linear transformations. For example, eigenvectors are described before linear transformations. This organization is pleasant to follow.
The interface in the electronic version is a selling point of the book. Every time a previous theorem or definition is invoked, the reader can click a link and view that previous theorem or definition without actually navigating to that page. Likewise, the book includes instruction on using the SageMath computer algebra system. The electronic version includes a direct interface to SageMath (through the SageMath Cell Server) which allows code to be run directly from the book.
I did not notice any grammatical or spelling errors.
I did not notice anything that was culturally insensitive or offensive.
Beezer's book includes all the expected topics in a first corse in linear algebra, and it also provides some review sections on set theory and read more
Beezer's book includes all the expected topics in a first corse in linear algebra, and it also provides some review sections on set theory and complex numbers. To place it in the broader world of linear algebra textbooks, this text is generally more algebraic and numeric than it is geometric: the section on Vector Operations, for instance, mentions that a vector might be thought of as "representing a point in three dimensions" and that one "can construct an arrow," but then finishes by saying "we will stick with the idea that a vector is just a list of numbers, in some particular order." The student gets early exposure to the axioms for vector spaces, which then link to the Proof Techniques section. The reference section at the end provides a list of notation, definitions, theorems. The online format does a nice job of providing an overall perspective on the course. The use of knols, for instance, lets a learner follow a reference without losing his/her place in the online text.
The book is mathematically accurate as far as I can tell, but there are also wonderul structural features of this book that ensure such accuracy. The content resides in a GitHub repo at https://github.com/rbeezer/fcla which makes it easy to submit edits (and indeed, to submit pull requests). The examples are supported by Sage code, which also makes mechanical errors unlikely in the presentation. As a globally-editable machine-assisted textbook, there are good reasons to believe it will remain accurate in future editions.
The incorporation of Sage certainly makes the content especially timely, especally with the tremendous excitement around https://cloud.sagemath.com/ In terms of longevity, the fact that the text of the book is stored in LaTeX and XML ensures that the text will be useful for a long time to come. Updates will be straightforward to implement. The book includes a lot of exercises.
In a definition, the word being defined is highlighted in bold. Examples are distinguished by a different background color. Sage code is supported with explanations (e.g., when thinking about Row Operations, the author explains that "[t]he copy() function, which is a general-purpose command, is a way to make a copy of a matrix before you make changes to it. " This sort of documentation is critical for guiding a student---perhaps without much python experince---to successly using the Sage environment to learn some mathematics). When making an argument, the author both names the property, and briefly recalls what it says in English: e.g., the author writes that "[s]ince every vector space must have a zero vector (Property Z), we always have a zero vector at our disposal." The text is friendly (literally about friends: the author writes "These will bring us back to the beginning of the course and our old friend, row operations.") without sacrificing rigor.
The book is consistent. Notation is presented at the end of the text, and used throughout. Objects are labeled with short acronyms and referred to throughout the book. Perhaps most important for consistency, the book uses a list of "archetypes" which are "typical examples of systems of equations, matrices and linear transformations" that have been crafted to "demonstrate the range of possibilities." By building the narrative around this small number of great examples, the book is pedagogically consistent.
The use of knols and "folding" does provide a degree of modularity: a student can be exploring one section and need not "open up" an example until they want to pursue that example. This format makes it very clear how the text is structured. I expect that instructors using this book would be using the material in the presented order, though, with the exception of perhaps pointing some students to the review sections at the end on complex numbers and sets.
Sections and theorems and the like are labeled with short acronyms instead of numbers; this appears a bit idiosyncratic at first, but it actually makes the text easier to read: the learner is more likely to assign meaning to "Theorem TSS" than they would "Theorem 17.42." As a grader, I much prefer it when my students provide clear names for the theorems they are invoking in their write-ups. In terms of organization, the book begins with concrete examples (e.g., column vectors) and then sections later provides "a formal definition of a vector space" which leads "to an extra increment of abstraction." This is a good way to provide a scaffold to more theoretical concerns, and is indicative of the thoughtful structure of the book overall.
The HTML interface is fantastic: the use of Knols lets a learner follow a link without losing the broader context. The math is rendered beautifully by MathJax. Examples and the like are "folded" so they do not distact the reader until he/she is ready to dig into the example. One thing that makes the book very useable is its use of the Sage cell server---the learner can use the interactive components of the textbook without having to install a local copy of Sage, which should make this book accessible by a broader number of people.
I have not noticed any grammatical errors. In terms of style, I would say that it is colloquial, friendly English. The material is certainly technical but there is a consultative, invitating tone behind the technical discussion. There is some concern to warn the reader about technical terms: e.g., the author wrtes "A final reminder: the terms [...] used in reference to vectors or matrices with real number entries are special cases of the terms."
There is a lot of great mathematical culture in the book, but perhaps not too many places where the book touches on "real-world examples" which might provide other places to touch on cultural issues.
This book covers a tiny bit more than I would normally cover in an introductory linear algebra class (due to its use of the complex numbers read more
This book covers a tiny bit more than I would normally cover in an introductory linear algebra class (due to its use of the complex numbers throughout), and omits nothing that I would normally cover. All subject areas address in the Table of Contents are covered thoroughly.
I found no accuracy issues in the text. Examples are worked out in full detail throughout the text, and at a first reading appeared to be error-free.
The content is as up-to-date as any introductory linear algebra textbook can reasonably be. The text includes some guidance on how to use Sage to help with calculations, but the book is written in such a way that it can be easily used without implementing Sage into the course.
I think that the text in this book is extremely clear, which is great for a first course in linear algebra. The book includes a few "one-liners" to help keep students engaged while reading, and I think that this is done really well!
This text is consistent in its terminology, both internally and globally.
The text is subdivided into small digestible chunks for students to read. The text is pretty self-referential, but the book is hyperlinked throughout. So it just takes one click for the reader to be directed to the definition, example, or section being referenced.
All of the material in the text follows from what has preceded it, so in that sense it is structured well. But my only very minor issue with this book is that some of the material is covered in what I would consider an "unusual order". Two big examples are: - eigenvalues and eigenvectors are covered before the notion of a linear transformation is defined. - The Gram-Schmidt procedure is introduced incredibly early in the text, before basic concepts like matrix operations, bases, dimension, or determinants. This is, of course, an opinionated issue though. Others may certainly like the ordering in this book better than what I would recommend. But the key is that the book is written so that one could easily "jump around" these parts without causing much confusion for the students. And due to the hyperlinks in the text, it is easy to navigate to the relevant sections. Also, a really nice touch is that there are 24 recurring examples throughout the text that the author calls "archetypes". These archetypes are all listed together at the end of the book, along with their description. I feel that this is great tool for students to easily be able to compare and contrast different types of examples.
I had no interface issues with this book. One interesting thing of note is that items are indexed using acronyms instead of numerically. For example, the fourth chapter is labeled "Chapter VS" instead of "Chapter 4" (and where VS is for vector spaces). I am not sure whether I like acronyms or numbers better, but it is all a moot point because of the hyperlinks used in the text. There is also a list of all acronyms used for definitions and theorems at the end of the book.
I found no grammatical errors in this textbook. It is very well written.
No portion of this text appeared to me to be culturally insensitive or offensive in any way, shape, or form.
Overall, I think that this textbook provides a great introduction to linear algebra! With such a great resource available to students for free, I do not see why I would ever force my students to purchase a different textbook in the future.
Table of Contents
- Chapter SLE Systems of Linear Equations
- Chapter V Vectors
- Chapter M Matrices
- Chapter VS Vector Spaces
- Chapter D Determinants
- Chapter E Eigenvalues
- Chapter LT Linear Transformations
- Chapter R Representations
- Chapter MD Matrix Decompositions
- Appendix CN Computation Notes
- Appendix P Preliminaries
- Appendix A Archetypes
- Appendix GFDL GNU Free Documentation License
- Part T Topics
- Section F Fields
- Section T Trace
- Section HP Hadamard Product
- Section VM Vandermonde Matrix
- Section PSM Positive Semi-definite Matrices
- Part A Applications
- Section CF Curve Fitting
- Section SAS Sharing A Secret
About the Book
A First Course in Linear Algebra is an introductory textbook aimed at college-level sophomores and juniors. Typically students will have taken calculus, but it is not a prerequisite. The book begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality. The final chapter covers matrix representations of linear transformations, through diagonalization, change of basis and Jordan canonical form. Determinants and eigenvalues are covered along the way.
A unique feature of this book is that chapters, sections and theorems are labeled rather than numbered. For example, the chapter on vectors is labeled "Chapter V" and the theorem that elementary matrices are nonsingular is labeled "Theorem EMN."
Another feature of this book is that it is designed to integrate SAGE, an open source alternative to mathematics software such as Matlab and Maple. The author includes a 45-minute video tutorial on SAGE and teaching linear algebra.
This textbook has been used in classes at: Centre for Excellence in Basic Sciences, Westmont College, University of Ottawa, Plymouth State University, University of Puget Sound, University of Notre Dame, Carleton University, Amherst College, Felician College, Southern Connecticut State University, Michigan Technological University, Mount Saint Mary College, University of Western Australia, Moorpark College, Pacific University, Colorado State University, Smith College, Wilbur Wright College, Central Washington U (Lynwood Center), St. Cloud State University, Miramar College, Loyola Marymount University.
About the Contributors
Robert A. Beezer is a Professor of Mathematics at the University of Puget Sound, where he has been on the faculty since 1984. He received a B.S. in Mathematics from the University of Santa Clara in 1978, a M.S. in Statistics from the University of Illinois at Urbana-Champaign in 1982 and a Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign in 1984. He teaches calculus, linear algebra and abstract algebra regularly, while his research interests include the applications of linear algebra to graph theory.