Linear algebra is fundamental to most mathematical branches, such as geometry and functional analysis. It is primary to pure as well as applied mathematics and opens to various subjects, including statistics, engineering, linear programming, and computer graphics. As a learner, you may wonder which books best teach linear algebra.
The best linear algebra book is ‘Linear Algebra: Theory, Intuition’ by Mike X Cohen. Unlike most texts, it covers all the main topics while incorporating the Python/MATLAB code in solving linear equations and computing problems. It explains ideas and theories clearly and comprehensively.
However, since the book is designed mainly for advanced learners, here’s a list of the top nine linear algebra books ranging from beginner to advanced level. This list is based on consultations with instructors and students. It is, however, not in any particular order of priority, so let’s begin!
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Best Linear Algebra Book for Beginners
The first three books in this list are geared towards students who have yet to be introduced to linear algebra.
1. Introduction to Linear Algebra – Gilbert Strang
This is one of those intelligent and discursive introductory texts. Gilbert Strang takes a unique approach to linear algebra. He focuses one eye on the theory part and the other on applications. He illustrates linear algebra concepts with a personal touch, concentrating mainly on matrices and linear equation systems.
The author also objectively touches on function spaces, calculus, wavelets, and the Fourier series. His presentation is not only straightforward but also informative, so much that even seasoned instructors find value in it.
He covers geometry and vector algebra, vector spaces, subspaces, orthogonality, determinants, linear equations, eigenvalues, and eigenvectors in the first six chapters.
The following three chapters focus on linear transformations, value decomposition, complex numbers, and complex matrices. The rest of the book addresses various cases of contemporary applications of linear algebra in computer programming, graphics, facial recognition, search engines, and so on.
This text provides sufficient learning material that thousands of scholars and instructors use to learn and teach linear algebra. In fact, it is the base coursebook for the Massachusetts Institute of Technology. Thanks to Strang’s splendid contribution to linear algebra, he has become an icon many young mathematics students look up to.
You can find his lectures on YouTube as well. Here’s one of them.
2. Linear Algebra and Its Applications – David C. Lay, Steven R., and Judith McDonald
This is a great beginner’s book, especially for self-studying students. It uses traditional texts to convey linear algebra ideas in easy, familiar, and solid terms. The text is legible and features several theoretical and practical examples and a handful of exercises to help test your knowledge. The surplus availability of solutions on the internet makes the self-learning process even smoother.
Whereas some basic principles of linear algebra, such as vector space and linear transformations, require a great deal of time and practice to assimilate, the authors of this book understand how crucial these concepts are to mastering the topic. They introduce them in a friendly and objective way, developing them bit by bit and explaining their practical applications to help learners understand them when presented in the abstract.
This book contains enormous information on linear algebra, even more than what most beginner books do than can be covered in just one semester. It has been revised over the years, and modern editions now center primarily on algorithms and applications.
3. Linear Algebra: Step by Step – Kuldeep Singh
Kuldeep Singh takes a comprehensive but straightforward step-by-step approach to explaining the basics of vector algebra and geometry, linear equations, and vector spaces. He gradually introduces abstract ideas along the way.
The text is so engaging that it often incorporates question and answer segments at the introduction of each method. That’s not all. At the end of each chapter, you’ll find additional practice questions derived from past examinations from different institutions. They are aimed at helping boost learners’ confidence and enable them to counter-check their progress. The best thing about these set questions is that you can easily find their solutions online for free.
At the beginning of each section, Kuldeep Singh includes the historical biographies of the key figures of linear algebra in relevance to the topics to be covered. This is to help you understand how most abstract lessons came to exist. The book also contains interviews with famous people who use linear algebra in real-life situations to enable you to relate to the application side of concepts
This book focuses on the core fields of linear algebra, including linear transformations, eigenvalues and eigenvectors, linear matrices, and linear equations.
Best Linear Books for Intermediate Learners
Once a student has gained an understanding of the basic concepts of linear algebra, they can move on to the next books on the list.
4. Linear Algebra Done Right 3rd Edition – Sheldon Axler
This book primarily targets third and fourth-year undergraduate mathematics students who study in North American universities but can be a valuable resource to any intermediate learner. It doesn’t require too much understanding of linear algebra to find value in it, but a grasp of basic concepts and a little mathematical maturity is necessary.
The author takes his students through a no-frills approach that opens a route to the primary purpose of algebra – understanding the structure of linear operators. The first few chapters focus on vector spaces, linear maps, polynomials, eigenvalues, and eigenvectors.
As the concepts become more abstract, the book expands to cover aspects such as complex vector spaces, inner-product spaces, and invariant subspaces. It simplifies concepts such as definitions and theorems using visual aids that are presented in colored boxes.
The book also features over 300 engaging exercises to help students understand the objects of linear algebra. Additionally, it offers numerous examples that explain the key notions of the subject. The proofs are also simplified for better grasping. Sheldon Axler’s piece of work is the type that shouldn’t be absent in every undergraduate’s library.
5. Basic Abstract Algebra: For Graduate Students and Advanced Undergraduates – Robert B. Ash
Robert B. Ash looks into the most basic algebraic structures, such as groups, maps, modules, rings, and fields. He attempts to express the elemental beauty of mathematics with clear reasoning behind every concept.
The author draws the methods from practical examples of real-life applications in physics, computer science, and engineering.
The text’s primary aim is to help students understand the subject through an informal or intuitive approach. It argues concepts informally in three ways:
- It provides concrete/numerical examples that point to the general case.
- Draws lessons from short surveys of large areas
- Paraphrases arguments intuitively, replacing lengthy unclear proofs.
Most higher-education institutions use this book as the primary graduate coursebook for studying abstract algebra. Furthermore, the book offers solutions to all the set problems, making it a valuable resource for self-studying learners.
6. A First Course in Linear Algebra – Robert Beezer
This text by Robert Beezer intersperses the standard topics of linear algebra with several others at an appropriate depth. The first set of chapters begins with topics like systems of linear equations, algebraic matrices, and vectors expressed in a well-organized text with prudently picked examples.
Once the students are well-equipped to transition into complex algebra, the chapters in the middle gradually introduce abstract vector spaces and other challenging topics. The last set of chapters focuses on more abstract aspects, such as orthonormal diagonalization.
The book contains sufficient details, proofs, and examples, but it has a poor labeling system that makes it hard to find references to other theorems. Despite having an appendix that sorts out such details by section, finding definitions is still a cumbersome task.
Regarding relevance, A First Course in Linear Algebra competes with other standard linear algebra texts, especially for instructors who want to implement programming/computation into their courses. The text, however, focuses on applications lightly, meaning you can’t really judge its cultural relevance.
Best Linear Algebra Books for Advanced Learners
Rounding out the list are three books that teach the advanced concepts of linear algebra as well as their applications.
7. Matrix Analysis and Applied Linear Algebra – Carl Meyer
What makes this book stand out is its focus on applications. It is a practical workbook that students find handy from a computational point of view. Carl Meyer strongly expresses ideas more precisely than in other undergraduate texts. In fact, he touches on some of the more contemporary topics that other authors of undergraduate books don’t and draws examples from practical situations to support them.
This text is a forthright mathematics resource that uses a novel approach to teach linear algebra. The author uses a new instructional method that establishes various questions and examples, ranging from the most basic to highly complex applications.
Meyer also discusses several features of linear equations using modern concepts and notations. Every theoretical topic is accompanied by examples that have been worked out in detail and expressed in a clear and informal writing style.
I found this book to be the most ideal for advanced self-learners since it comes with a solutions manual for all set problems. It gives students the privilege to further their insight at the end of each section with over 240 examples and 650 exercises.
By including a CD-ROM that contains a searchable copy of the whole book, detailed information about topics, references, and the history of linear algebra and computing, I can say that Carl Meyer’s book is insightful, loveable, and exciting to read.
8. Advanced Linear Algebra – Steven Roman
This textbook exhaustively introduces linear algebra to advanced undergraduate learners. It, however, requires prior knowledge of elementary features of matrices.
It is also appropriate for students who previously studied linear algebra at lower levels since it covers elementary vector spaces and linear transformations more briskly. This is done in the first three chapters of the text.
In chapters 4,5 and 6, the author discusses modules and emphasizes a comparison between their properties and those of vectors. He also talks about several theories associated with modules, which he used to establish a theorem for finitely created modules.
He discusses those linear operators in the 7th and 8th chapters, while the rest of the book focuses on real and complex inner products. It goes ahead to touch extensively on tensor products, metric vector spaces, and umbral calculus.
In this book, Steven Roman takes a direct approach to teaching. He expresses linear algebra concepts in a concrete way designed to equip the learner with the basics and techniques of formal mathematics.
At the end of each section are three reading questions intended to help learners gauge their learning progress. The book also features several examples and real-life applications of linear algebra that prove theorems carefully.
9. Linear Algebra: Theory, Intuition – Mike X. Cohen (Best Overall)
This is another unique text that comprehensively combines explanations, examples, visualizations, and code (Python and MATLAB) to demonstrate linear algebra concepts. It’s everything you need if you are looking to implement linear algebra concepts on computers.
In this book, Mike Cohen uses an authentic learning strategy by drawing several diverse explanations to stress a specific concept. He enhances the geometric aspect of linear algebra using graphs and other visualization methods.
And since we no longer solve mathematical problems manually, this book teaches you how to implement MATLAB and Python when solving them. Codes are crucial since they help you understand how computing’s linear algebra concepts are applied.
In addition, this 600-page text contains all the essential topics of linear algebra, from basic to advanced levels. It touches on vectors and spaces,least-square projections, matrix multiplications, eigendecomposition, and singular-value decomposition.
This text also heavily leans towards the current application features of linear algebra as well as matrix analysis. While the author understands that mathematics isn’t a spectator sport, the book comes with a blend of hand-solved problems and more cutting-edge code exercises to help you learn and master the subject.
Final Thoughts
After going through this list of the nine best algebra books, you’ve hopefully expanded your knowledge of linear algebra and mathematics in general. Before you buy any book on this list, you must ensure that they meet your required prerequisites.
I tried to make everything easier for you by classifying the books according to their levels, so all you have to do is select the text within your level of learning and start studying it.
If you’re looking to get into calculus next, make sure to check out this article about the 14 best calculus books.
