Learn Linear Algebra: Best Textbooks and Study Roadmap
Table of Contents
What Is the Best Way to Learn Linear Algebra?
The most effective path through linear algebra emphasizes vector spaces and linear transformations rather than rote matrix computation. Start with proof techniques and basic matrix operations, then work through a rigorous introductory text such as Friedberg, Insel, and Spence’s Linear Algebra, before moving to specialized follow-up books based on your goals.
Key Takeaways
- Friedberg, Insel, and Spence’s Linear Algebra (4th edition) is recommended as the primary introductory textbook, covering vector spaces through canonical forms.
- Sheldon Axler’s Linear Algebra Done Right delays determinants until its final chapter, so it works better as a supplementary text than a first course.
- Sergei Treil’s Linear Algebra Done Wrong is freely available online at math.brown.edu and covers nontraditional topics including tensors.
- Carl D. Meyer’s Matrix Analysis and Applied Linear Algebra requires no prior linear algebra background and focuses on numerical and applied computation.
- Two proof prerequisites are needed before starting: familiarity with proof techniques and basic matrix/determinant operations such as 2×2 and 3×3 examples.
- Alan Macdonald’s Linear and Geometric Algebra and the Dorst, Fontijne, and Mann text both extend standard linear algebra into geometric algebra (Clifford algebras) without requiring additional prerequisites.
What Prerequisites Are Needed Before Studying Linear Algebra?
Beyond standard high-school algebra, geometry, and trigonometry, calculus is not a formal prerequisite for linear algebra. Two specific prerequisites matter more: familiarity with proof techniques, and basic exposure to matrices and determinants.
Recommended Proof Books
Three proof-based textbooks introduce the set-theory notation and reasoning skills needed for abstract and linear algebra:
- Daniel Velleman, How to Prove It: A Structured Approach (2nd edition) — available on Amazon
- Ethan Bloch, Proofs and Fundamentals: A First Course in Abstract Mathematics — available on Amazon
- Richard Hammack, The Book of Proof — free online edition
Matrix and Determinant Basics
Basic matrix and determinant material is typically covered in high-school or introductory college courses. Useful resources include Serge Lang’s Basic Mathematics, OpenStax’s free College Algebra textbook, and Khan Academy’s linear algebra matrix transformations unit.
Complete mastery of matrices and determinants is not required before starting a formal linear algebra course. Familiarity with 2×2 and 3×3 examples, computing matrix inverses, and solving linear systems is enough, since introductory linear algebra texts build the full theory from there.
What Does an Introductory Linear Algebra Course Cover?
A proper linear algebra course emphasizes vector spaces and linear transformations between them, then rigorously develops n×n matrix and determinant theory and applies these ideas in both geometric and algebraic contexts.
Friedberg, Insel, and Spence’s Linear Algebra (4th edition) is recommended as the introductory textbook for this approach. It is available at Amazon.
Topics Covered in Friedberg, Insel, and Spence
| Topic | Key concepts included |
|---|---|
| Vector spaces | Linear dependence, subspaces, bases, dimension |
| Linear transformations and matrices | Rank, isomorphisms, change-of-basis matrices, dual spaces |
| Determinants | Rigorous proofs of determinant properties |
| Diagonalization | Eigenvalues, Markov chains, the Cayley–Hamilton theorem |
| Inner product spaces | Gram–Schmidt process, spectral theorem, singular value decomposition |
| Canonical forms | Jordan canonical form, rational canonical form |
An explanation of the Cayley–Hamilton theorem using Python’s SymPy module is available for readers who want a computational perspective on diagonalization.
Which Follow-Up Linear Algebra Textbooks Are Worth Reading?
Linear Algebra Done Right by Sheldon Axler
Sheldon Axler’s Linear Algebra Done Right contains elegant proofs and useful perspectives, including a particularly instructive existence-of-eigenvector argument. It is not recommended as a first introduction because it avoids determinants until a very abstract final chapter, so a more conventional introduction should come first. It is available at Amazon. Prerequisite: a course on linear algebra such as Friedberg, Insel, and Spence.
Linear Algebra Done Wrong by Sergei Treil
Sergei Treil’s Linear Algebra Done Wrong is terse and works better as a second book than a first one. It covers nontraditional topics such as tensors and is freely available online at Brown University’s mathematics department site. Prerequisite: a course on linear algebra such as Friedberg, Insel, and Spence.
Matrix Analysis and Applied Linear Algebra by Carl D. Meyer
Carl D. Meyer’s Matrix Analysis and Applied Linear Algebra focuses on the applied and numerical side of linear algebra, addressing practical issues such as misconceptions about determinants and near-singularity that theoretical texts often skip. It is available at Amazon. No prerequisites are required.
3000 Solved Problems in Linear Algebra by Seymour Lipschutz
Seymour Lipschutz’s 3000 Solved Problems in Linear Algebra is filled with computational problems and some proof-based problems, useful for readers who want extensive practice. It is available at Amazon. No prerequisites are required.
Linear and Geometric Algebra by Alan Macdonald
Alan Macdonald’s Linear and Geometric Algebra covers both linear algebra and geometric algebra (Clifford algebras), which provides a unified viewpoint for quaternions and forms and offers an alternative interpretation of the determinant that introductory courses often omit. It is available at Amazon. No prerequisites are required.
Geometric Algebra for Computer Science by Dorst, Fontijne, and Mann
Despite its title, this book by Leo Dorst, Daniel Fontijne, and Stephen Mann is a substantial treatment of geometric algebra that provides intuition useful in both physics and computer science, covering more ground than Macdonald’s book. It is available at Amazon. No prerequisites are required.
Linear Algebra via Exterior Products by Sergei Winitzki
Sergei Winitzki presents a basis-free approach to linear algebra that emphasizes the wedge product rather than coordinate computations, making it a good companion to geometric algebra and a solid sequel to a standard linear algebra course. It is available at Amazon. Prerequisite: a course in linear algebra such as Friedberg, Insel, and Spence.
Advanced Linear Algebra by Steven Roman
Steven Roman’s Advanced Linear Algebra offers a graduate-level perspective, covering modules, Hilbert spaces, and umbral calculus. It requires mathematical maturity, including background in analysis and abstract algebra plus prior linear algebra study. It is available at Amazon. Prerequisites: analysis, abstract algebra, and linear algebra.
Frequently Asked Questions
Do I need calculus before studying linear algebra?
No. Calculus is not a formal prerequisite for linear algebra. Standard high-school algebra, geometry, and trigonometry are sufficient, along with familiarity with proof techniques and basic matrix and determinant operations.
What is the best first textbook for linear algebra?
Friedberg, Insel, and Spence’s Linear Algebra (4th edition) is recommended as the first textbook because it rigorously develops vector spaces, linear transformations, determinants, diagonalization, inner product spaces, and canonical forms in a modern, structured way.
Why isn’t Axler’s Linear Algebra Done Right recommended as a first book?
Sheldon Axler’s Linear Algebra Done Right delays determinants until a very abstract final chapter, and determinants matter for many applications. A more conventional introduction, such as Friedberg, Insel, and Spence, should come first, with Axler used as a supplement afterward.
Is there a free linear algebra textbook available online?
Yes. Sergei Treil’s Linear Algebra Done Wrong is freely available online through Brown University’s mathematics department. It is terse and covers nontraditional topics such as tensors, making it better suited as a second book than a first introduction.
What is geometric algebra and how does it relate to linear algebra?
Geometric algebra, also called Clifford algebra, extends linear algebra concepts to provide a unified viewpoint for quaternions, forms, and related structures. Alan Macdonald’s Linear and Geometric Algebra and the Dorst, Fontijne, and Mann textbook both cover this material and require no additional prerequisites beyond general mathematical background.
Which book should I use if I need lots of practice problems?
Seymour Lipschutz’s 3000 Solved Problems in Linear Algebra is filled with computational problems and some proof-based problems, making it a strong choice for readers who want extensive practice alongside a standard textbook.
Advanced education and experience with mathematics
I think Hoffman and Kunze is a bit abstract for most beginners, having been aimed at a junior level math major at MIT.
after reading some introductory book you might possibly gain something from my notes. In the first 3 pages I summarize the entire content of the most advanced parts of most books, the jordan and rational canonical normal forms. it is free of course. i also like shilov very much, but have not really worked through it. a lot of people, including me, like friedberg insel and spence. but they just don't tell you what is behind the results as i try to do, namely they don't mention the main idea for understanding linear transformations, i.e. minimal polynomials.
http://alpha.math.uga.edu/~roy/laprimexp.pdf
what are your thoughts on Kenneth M Hoffman's Linear Algebra textbook?
This surely fits the list: https://www.lem.ma/Made by: http://drexel.edu/coas/faculty-research/faculty-directory/PavelGrinfeld/
“Thank you micromass for the second insight this month that meets me where I’m at (the other was the one about advanced mathematics for high school students)!
As someone who is currently attempting to self-study linear algebra (albeit very slowly), this post helped me to see what I can expect to know once I’m done (by what micromass says the other books cover), especially since I’m using a fairly “mathematically pure” book to study from that doesn’t give much motivation the whole subject.
Since it didn’t make it on micromass’ list, I’ll go ahead and mention the book I’m using for anyone who is interested. It’s by Shilov, just called “[URL=’http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X’]Linear Algebra[/URL].” It starts from determinants, which I hear is a different approach than most books take. The second chapter is on linear spaces though, so it seems that it still gets to “the point” rather quickly. I find it excellent, and perhaps it should be considered for someone who is looking into the subject.
Although I will point out that I have no prior experience with linear algebra and have looked at no other texts, so please take micromass’ textbook advice over mine (if only this post had come out before I started to get into the book :H).”
Don’t worry, Shilov is an excellent book. You really can’t go wrong with it. I personally wouldn’t recommend it because it does determinants in the beginning, which I find a fairly unintuitive and perhaps too abstract approach. Also, it never really says what a determinant is geometrically (as far as I recall). But if you like it, it’s a nice book.
Thank you micromass for the second insight this month that meets me where I’m at (the other was the one about advanced mathematics for high school students)!
As someone who is currently attempting to self-study linear algebra (albeit very slowly), this post helped me to see what I can expect to know once I’m done (by what micromass says the other books cover), especially since I’m using a fairly “mathematically pure” book to study from that doesn’t give much motivation the whole subject.
Since it didn’t make it on micromass’ list, I’ll go ahead and mention the book I’m using for anyone who is interested. It’s by Shilov, just called “[URL=’http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X’]Linear Algebra[/URL].” It starts from determinants, which I hear is a different approach than most books take. The second chapter is on linear spaces though, so it seems that it still gets to “the point” rather quickly. I find it excellent, and perhaps it should be considered for someone who is looking into the subject.
Although I will point out that I have no prior experience with linear algebra and have looked at no other texts, so please take micromass’ textbook advice over mine (if only this post had come out before I started to get into the book :H).
One thing about the Macdonald book is how surprisingly small it is (204 pages) for the amount of content it seems to cover. This is mainly for 2 reasons: (1) it handles worked exercises in a cool way and (2) he doesn’t devote space to learning what he calls algorithms (e.g. the mechanistic cookbook recipe for row reduction, etc.)
Regarding worked exercises, the trick is he has you do them! Almost every page has a couple of small exercises that relate to the text you just read. They really make you engage with the content as you go in a neat way that I haven’t seen before. Sometimes you’ll want a little scratch pad and a pencil to work it out and other times it’ll be something simple that you can work out in your head like “what happens if you set t = 0 or 1?” and then you have an aha moment as you realize it simplifies to something you’ve seen before. This is quite rewarding as opposed to being given the same information in a paragraph.
Regarding algorithms, an example is matrix inversion – he goes through the concept and applications of it, thereafter using it throughout the book but he does not devote space to building up the detailed recipe for mechanistically computing one by hand. Same goes for row reduction, determinants, eigenstuff, etc. In the Preface he argues that the recipes are not needed for theoretical development, and no one solves them by hand anymore anyways except as exercises in Linear Algebra textbooks.
Anyone intending to tackle both the Linear Algebra Insight and the Intro Analysis Insight, will probably notice that there is some overlap between the two. Micromass was kind enough to provide an efficient way to navigate through them, which he gave permission to repost here:
“So if you’re doing both of them, then I would recommend:
Do Bloch Analysis and MacDonald in parallel.
Then after Bloch do Hubbard, and after MacDonald do Axler.
This way you’ll get everything without too much repetition. MacDonald will teach you the basics of LA (vector spaces, linear transformations), but will also do geometric algebra. Hubbard will repeat the basics but not from a point of view of analysis. And Axler will do things in the most rigorous light. Avoiding determinants in Axler is not a problem since Hubbard and MacDonald cover those. What do you think? It is possible to do Treil instead of Axler if you prefer Treil, but it’s really up to you.”
Great list – thanks for putting this together!
The author of Linear and Geometric Algebra, Alan Macdonald, has put together some youtube videos in support of his book.
[URL]https://www.youtube.com/c/AlanMacdonald1/playlists[/URL]
One thing about the Macdonald book is how surprisingly small it is (204 pages) for the amount of content it seems to cover. This is mainly for 2 reasons: (1) it handles worked exercises in a cool way and (2) he doesn't devote space to learning what he calls algorithms (e.g. the mechanistic cookbook recipe for row reduction, etc.)Regarding worked exercises, the trick is he has you do them! Almost every page has a couple of small exercises that relate to the text you just read. They really make you engage with the content as you go in a neat way that I haven't seen before. Sometimes you'll want a little scratch pad and a pencil to work it out and other times it'll be something simple that you can work out in your head like "what happens if you set t = 0 or 1?" and then you have an aha moment as you realize it simplifies to something you've seen before. This is quite rewarding as opposed to being given the same information in a paragraph.Regarding algorithms, an example is matrix inversion – he goes through the concept and applications of it, thereafter using it throughout the book but he does not devote space to building up the detailed recipe for mechanistically computing one by hand. Same goes for row reduction, determinants, eigenstuff, etc. In the Preface he argues that the recipes are not needed for theoretical development, and no one solves them by hand anymore anyways except as exercises in Linear Algebra textbooks.
Anyone intending to tackle both the Linear Algebra Insight and the Intro Analysis Insight, will probably notice that there is some overlap between the two. Micromass was kind enough to provide an efficient way to navigate through them, which he gave permission to repost here:"So if you're doing both of them, then I would recommend:Do Bloch Analysis and MacDonald in parallel.Then after Bloch do Hubbard, and after MacDonald do Axler.This way you'll get everything without too much repetition. MacDonald will teach you the basics of LA (vector spaces, linear transformations), but will also do geometric algebra. Hubbard will repeat the basics but not from a point of view of analysis. And Axler will do things in the most rigorous light. Avoiding determinants in Axler is not a problem since Hubbard and MacDonald cover those. What do you think? It is possible to do Treil instead of Axler if you prefer Treil, but it's really up to you."
Great list – thanks for putting this together!The author of Linear and Geometric Algebra, Alan Macdonald, has put together some youtube videos in support of his book.https://www.youtube.com/c/AlanMacdonald1/playlists