Best linear algebra books

I would suggest starting with a basic level textbook and then put more effort on one or two intermediate level textbooks. Advanced level books may not be a good source for study.

Basic

Introduction to Linear Algebra 6th Edition by Gilbert Strang

This is an erudite and discursive introduction to linear algebra, weighted heavily toward matrices and systems of linear equations. The author has an expansive view of linear algebra, and from time to time draws in some calculus, Fourier series, wavelets, and function spaces, but the approach is always very concrete. The book doesn’t skimp on the techniques of linear algebra, and there are seemingly endless examples of LU-decomposition and other numeric work, as well as a fairly extensive Chapter 9 on numerical methods. But the book also does a good job of moving up and down between various levels of abstraction, according to which level makes the problem at hand easier to comprehend, and geometrical examples and rotations play an important role in the exposition.

This book is the text for Massachusetts Institute of Technology’s Linear Algebra course 18.06, whose goals are “using matrices and also understanding them.” There’s enough material in the book for a year-long course, and the MIT course covers primarily the first seven chapters. There is a great deal of free supplemental material available on the MIT web site for the course, including videos of Strang’s lectures in the course, interactive computer demonstrations, and past exams and problem sets. The present book is complete in itself, but many students will appreciate the additional resources. This book is also good for self-study as there is a solution munual for it.

Linear Algebra and Its Applications 5th Edition by David C. Lay, Steven R. Lay, Judi J. McDonald

(My personal pick for a first course) This is a no-frills textbook for a one-semester course in linear algebra that focuses very heavily on algorithms and applications. Despite the no-frills approach, it is still a long book and it covers everything that would normally be in an introductory course, and a lot that would not be. There is much more material here than could be covered in a semester.

The breadth of applications is especially impressive: although most of them use extremely simplified models of the thing being studied, they really do give you a good understanding of how linear algebra is used in practice. The applications cover many areas of science, business, and engineering, with a lot of dynamical systems examples. There are also many notes on numerical considerations. None of this goes into enough depth to make you an expert (that would be impossible in a one-semester introductory course), or even able to tackle such applications on your own, but it does give you a good understanding of how linear algebra is used and why it is important.

No bullshit guide to linear algebra by Ivan Savov

This is the only book I’ve found that can take you from absolute beginner to confident intermediate in just a few weeks. How? First, the title is the truth. There is not an ounce of filler in this whole book. Every section jumps right into the subject and walks you through step-by-step until you understand the whole thing. Nothing is left out, and everything is explained in clear, readable, conversational English. And there are none of these “guess my answer” games either. Every section is loaded with practice problems so you can check your learning, and the author doesn’t cop out and only give the answers to a few of them. Every single problem is included in the answer key. He also has taken the time to point you to useful videos on YouTube that are relevant to the section’s material, which is a huge time saver. Plus, he has several chapters showing real-life applications.

This book is also good for self-study and has brief solutions for exercises.

Linear Algebra: Step by Step by Kuldeep Singh

If you desire to learn something useful and general about Linear Algebra, this book is where you should(must) begin. It provides proofs, worked examples, diagrams, and plenty of exercises with brief solutions at the end of the book. If desired, a website is provided for obtaining detailed solutions.

This book reminds one of the Programmed Instruction books of the 50s, 60s, 70s, and 80s where one was forced to learn by having to work through the text. However, it is not a Programmed Instruction book!

As you progress through this book, it becomes clear that you are gaining practical math based skills. You are confident that subjects requiring theoretical and/or practical exposure to Linear Algebra will no longer be intractable.

This is an indisputable five star book. You feel confident in applying what has been learned to any problem requiring knowledge of Linear Algebra.

By the way the interspersed autobiographies are motivating.

This book is also good for self-study and has brief solutions for exercises.

Intermediate

Linear Algebra Done Right 3rd Edition by Sheldon Axler

(My personal pick for a second course) One of my favoraite books on Linear Algebra. This book can be thought of as a very pure-math version of linear algebra, with no applications and hardly any work on matrices, determinants, or systems of linear equations. Instead it focuses on linear operators, primarily in finite-dimensional spaces but in many cases for general vector spaces. Solutions can be found here.

The book is not as bold as its title indicates; “done right” refers to the very technical device of avoiding determinants. But avoidance does have the useful effect of forcing you to think directly in terms of vectors and operators and not dive into a pile of calculations. Axler has come up with some very slick proofs of things that normally require a lot of grinding away, and that in itself makes the book interesting for mathematicians. The book is also very clearly written and fairly leisurely. It is printed in full color and has lots of sidebars and incidental graphics. But it is quite affordable.

Linear Algebra 2nd Edition by Kenneth Hoffman and Ray Kunze

(My personal pick for a second course) One of my favoraite books on Linear Algebra. Though old and classical, it is one of the best linear algebra books for math major students. Solutions can be found here.

Regarding Hoffman-Kunze, suffice it to say that all undergraduate-level material is done the right way (and then some), meaning that everything is proved, very carefully and with no compromises, and material is dealt with that is most often introduced no earlier than in graduate algebra, or possibly in an honors course in advanced linear algebra.

I want to lament in this connection that over recent decades a shift has occurred regarding linear algebra: things once covered in lower division work are now part of the standard graduate course, and accordingly erstwhile solidly undergraduate linear algebra, e.g., the full discussion of the eigenvalue problem leading to the Jordan canonical form, have fallen off the undergraduate table. The result is that this fantastic book is not usable in most undergraduate linear algebra courses. But it is an unsurpassed text for highly gifted kids who intend to do graduate school right and then go on to do real mathematics.

Finite-Dimensional Vector Spaces 2nd Edition by Paul Halmos

This is a classic but still useful introduction to modern linear algebra. It is primarily about linear transformations, and despite the title most of the theorems and proofs work for arbitrary vector spaces.

The book does start from the beginning and assumes no prior knowledge of the subject. It’s also extremely well-written and logical, with short and elegant proofs. But it has no pictures, no worked examples, no discussion of algorithms or numerical methods, and very little motivation. It’s really hard to know where you’re going if you’ve never studied linear algebra before, so it’s extremely helpful to have already had a course in matrices and linear algebra in RnRn. Most of the book is structured as “define something; prove a few theorems about the something; repeat with a new something.” After 55 pages of this we find the dismaying admission “We come now to the objects that really make vector spaces interesting.”

The exercises are very good, and are a mixture of proof questions and concrete examples. The book ends with a few applications to analysis (Halmos’s true interest in the subject) and a brief summary of what is needed to extend this theory to Hilbert spaces.

Linear Algebra 5th Edition by Friedberg, Insel, Spence

This seems to be the standard choice for honors undergraduate courses in the US these days. It is more challenging than the usual computational type introductions to linear algebra. If you want something more applied and less theoretical than the above three books, this is the best linear algebra textbook for you.

Pros:

Great for self-study. If you work the problems in each section, you’ll have a much better grasp of the topic as a whole. There is no official answer key, but many of the problems that look tough are not. (Do not skip the quotient space problems!)

Elegantly prepares its readers for upcoming topics to the point where important results begin to seem obvious. This book can make you feel smarter than you are, because there are no ugly surprises in the exercise sections and each new section builds from its predecessors.

Errata provided online.

Problems which provide results used later in the textbook are (usually) marked.

Advanced

Advanced Linear Algebra 3rd Edition by Steven Roman

(My personal pick for reference) This is a formidable volume, a compendium of linear algebra theory, classical and modern, intended for “the graduate or advanced undergraduate student.” (I have not had the privilege of teaching undergraduates who could handle this text.) After a concise (30-page) treatment of set theory and basic algebraic structures, the author embarks on a two-chapter whirlwind tour of introductory linear algebra, including an optional discussion of topological vector spaces. Following this, there are several chapters of module theory, leading to structure theorems for finite-dimensional linear operators. The last parts of the “Basic Linear Algebra” section of the book are devoted to real and complex inner product spaces and the structure of normal operators.

But this is only slightly more than half of this book’s contents! The last part of the volume is devoted to various “topics” such as Hilbert spaces, tensor products (including a treatment of the determinant as an antisymmetric n-linear form), affine geometry, QR and singular value decompositions, and the umbral calculus (“the first time that this subject has appeared in a true textbook”).

The development of the subject is elegant, positively Bourbakian. The proofs are neat, whereas the examples are sparse and would have to be supplemented by the instructor. The exercise sets are good, with occasional hints given for the solution of trickier problems. Appropriate undergraduate prerequisite texts would seem to be Jacobson’s Basic Algebra I and the classic text (still in print) of Hoffman and Kunze.

Linear Algebra in Action by Harry Dym

These are the standard references at the advanced level, and deservedly so. The coverage is comprehensive, but I wouldn’t want to try to learn linear algebra from them.

Reviews are from MAA and Amazon.