Beginner’s Review of College Level Linear Algebra Books

A little note: this is an old draft that I wrote more than a year ago, that somehow didn’t get published. I forgot why I didn’t publish it, maybe it was not completely finished. But rereading it now, it looks good enough for me, so I’m publishing it as is.

But.. before you buy any of the books here, you may want to wait a little bit because I’m writing another article about a very good video resource on the topic.

P.S. Enggak tahu juga kenapa (waktu itu) saya nulisnya pakai bahasa Inggris yah..

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Sometime in your machine learning learning journey, you will probably want to learn deeper about linear algebra since it is the language of machine learning. You start searching for the best book to read. There are a lot of different recommendations out there. The problem is, they all seem to be good books, so which one to choose? After all, you will be investing a lot of time reading just one book, and probably there is no time to read more than one, so you want to make the best out of it.

The answer is, of course, it depends. For the most part, it depends on the level of understanding you currently have. What people recommend may not be best suited for you, for this reason.

And the same applies this article.

So let me start by stating who this recommendation is for.

This article is intended for readers who are, like me, looking for the first college level linear algebra book to read. You probably have had some exposure to linear algebra before (such as knowing what matrices are and some basic operations to them), but otherwise had no prior formal exposure to it.

Let’s have some tests. These are common terms in college level linear algebra books. Count how many of them you can say they are comfortably make sense to you:

  1. matrix multiplication
  2. trace
  3. norm
  4. linearly independent
  1. basis
  2. subspace
  3. rank
  1. determinant
  2. hermitian
  3. eigenvalues

If you know about three or less, then you’re probably a beginner like me and this article is for you. If you know about five or less, then you may still benefit from reading this review since I feel that it doesn’t hurt to re-learn from the beginning to make sure that your understanding is “proper”. If you know more, then frankly I don’t know what to recommend; you know better than me.

So here are the books that I reviewed.

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The Books

The Manga Guide to Linear Algebra – Shin Takahashi (2012)

The Manga Guide to Linear AlgebraPaperback: 264 pages
Publisher: No Starch Press; 1 edition (May 1, 2012)
Language: English
ISBN-10: 1593274130

Understanding linear algebra by reading a comic certainly sounds like an interesting thing to do. Naturally this is the first book that I read.

Despite the, well, comical nature of the book, this actually is a very good material! It starts off from the very very beginning of linear algebra, the fundamentals, which tells stories about things such as number systems, equivalence, set theory, functions, and so on. Even matrix is only explained in third chapter, page 63. Then it builds up to heavier subjects such as determinants, calculating inverse matrices, solving linear equations using both Gaussian and Cramer rules, linear independence, bases, ranks, eigenvalues and eigenvectors. Basically all the stuffs that you expect from college level algebra book! But in a comic! Amazing!

I intended to read this as a warm up reading before taking on the heavier books. But after finished reading it, it feels that I already have my bases covered. I may not need to read other books!

But this “book” is not perfect though. While it does a good job in explaining a lot of important stuffs in the beginning, the explanations of the more advanced topics towards the end are pretty difficult to understand. I found sections about geometric representation of vector, subspace and dimension, and calculating kernels, are difficult to understand that I needed to seek explanation elsewhere to try to make sense of them.

Conclusion: if you need the very first introduction to linear algebra “book”, this is the best. It gets things understood quickly, with enough examples, and you get entertained at the same time.

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Linear Algebra 3rd Ed – Jim Hefferon (2017)

Linear Algebra Jim HefferonPaperback: 508 pages
Publisher: Orthogonal Publishing L3C; 3 edition (January 1, 2017)
Language: English
ISBN-10: 1944325034

This book is free (!), you can get it from the author’s website.

I find this book to be easy to read. On every subject, it starts off with a problem and then gently guides the reader to particular ways to solve it. It starts from the very beginning on why LA is needed, then guide the reader to matrix representation of the problem and explains why the matrix form is better. Contrast this with many other books that directly jumps into explaining matrix representation without proper introduction on what they were invented for.

The explanation about subjects are exhaustive, with a lot of examples and the proofs when they are needed.

Conclusion: very good introduction to LA book. This is the one that I will probably read.

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Linear Algebra and Its Applications (4th ed.) – Gilbert Strang (2006)

Linear Algebra and Its Applications StrangHardcover: 487 pages
Publisher: Brooks Cole; 4th edition (2006)
Language: English
ISBN-10: 0030105676

This book is recommended less heavily by people than the other Strang’s book (Introduction to Linear Algebra, see below). As with other introductory books to LA, it starts with the central problem of LA, i.e. solving linear equations.

It looks like the book assumes the reader has had some decent exposure to LA before, for the fact that the first page of chapter 1 already talks about determinants as a way to solve equations without giving explanations about it.

I also found the explanations in this book to be short and direct. This is good if you want to cover subjects quickly, but the risk is if you get stuck at something you may need to find the explanations elsewhere.

Conclusion: not to be used as the very first book on LA. Subjects are taught in short and concise ways (i.e. less explanations), which may or may not suit you.

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Introduction to Linear Algebra 4th Ed – Gilbert Strang (2009)

Introduction to Linear Algebra StrangHardcover: 584 pages
Publisher: Wellesley Cambridge Press; 4 edition (February 10, 2009)
Language: English
ISBN-10: 0980232716

This book is heavily recommended by a lot of people, usually for the fact that the author is a professor at MIT and he teaches an online linear algebra course at MIT and this book is used as the reference for that course.

I find this book to be a bit difficult to read. It feels like this book is for people who have been exposed to linear algebra before. For example the very first paragraph on Chapter 1 jumps straight into vectors without any explanations of what they are.

Conclusion: if you have some exposure to LA before, this might be one of the best LA books for you to learn LA deeper as it has been recommended by a lot of people.

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Linear Algebra 3rd Ed – Serge Lang (2004)

Linear Algebra Serge LangHardcover: 285 pages
Publisher: Springer; 3rd edition (March 9, 2004)
Language: English
ISBN-10: 0387964126

As the author says this book is intended for students with some exposure to LA before. For first exposure to LA, he has another book, Introduction to Linear Algebra. Unfortunately I was not able to acquire that book for reviewing purpose.

This book emphasis is on various structure theorems such as eigenvalues and eigenvectors, symmetric, hermitian and unitary operators, triangulation of matrices and linear maps, Jordan canonical form; convex sets and the Krein-Milman theorem.

Conclusion: not for beginning students

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Linear Algebra 4th Ed (Schaum’s Outline) – Lipschutz, Lipson (2009)

Series: Schaum’s Outline Series
Paperback: 432 pages
Publisher: McGraw-Hill; 4 edition (August 26, 2008)
Language: English
ISBN-10: 007154352X

This book aims to present an introduction to linear algebra which will be found helpful to all readers regardless of their fields of specification.

I find this book to be suitable for beginners. It starts with with basic concepts about vector and builds up the operations from there. The way the author explains a subject is concise, with a lot of examples and the necessary proof when it is needed.

Just as the title says this is an outline, topics are presented like an outline, i.e. the author just provides the definitions of some topic and then quickly followed by another (usually related) without too many motivating stories behind them. For example, he may explains dot product operation and orthogonality in just two short paragraphs, without any motivating “stories” behind them.

As a side note, many reviewers point out that this book may contain many errors.

Conclusion: okay for beginners, but probably best used as review book.

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Linear Algebra Done Right 3rd Ed – Sheldon Axler (2015)

Series: Undergraduate Texts in Mathematics
Hardcover: 340 pages
Publisher: Springer; 3rd ed. 2015 edition (November 6, 2014)
Language: English
ISBN-10: 3319110799

This book is another book that is heavily recommended by the good people on the Internet. The preface says this book is more suited for students on their second exposure to LA, not the first. However, it also says that it starts from the beginning of the subject, assuming no knowledge of LA.

The first chapter starts with vector spaces, and the reader is assumed to have already been familiar with basic properties of the set R of real numbers, because the chapter will explain about complex numbers. Already this may be too advanced for beginners.

Conclusion: not to be used as the first LA book

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Linear Algebra Done Wrong – Sergei Treil (2017)

213 pages
Published 2004

At 213 pages, this book is significantly shorter than the others. I find the explanations to be practical and probably less formal/theoretical than the others. They are concise and to the point.

I haven’t read this book enough to make conclusions, but I think I would probably prefer books that are more “formal” so that I would be able to understand something by their “proper” (formal) definitions.

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