Linear algebra done right

Last night I was looking into what to read after or along with 3Blue1Brown's series of Linear algebra videos [1]

The contenders seems to be:

- Linear Algebra Done Right - Sheldon Axler

- Liner Algebra Done Wrong - Sergei Treil

- Introduction to Linea Algebra - Gilbert Strang

- Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares by Stephen Boyd and Lieven Vandenberghe

[1] https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

Previously on Hacker News:

Linear Algebra Done Right 58 points, July 2023, 4 comments https://news.ycombinator.com/item?id=36576114

Linear Algebra Done Right – 4th Edition, 631 points, Oct 2023, 294 comments https://news.ycombinator.com/item?id=38060159

Linear Algebra Done Right [pdf], 85 points, Sept 2024, 39 comments https://news.ycombinator.com/item?id=41416799

For those who find Linear Algebra Done Right too much to start with, and those who don't get why Strang starts with matrices, I can't recommend more "The dark art of linear algebra" read this first. With this you can then tackle every other book on the topic more easily

This is supposedly based on Sheldon Axler's earlier and shorter paper "Down With Determinants!" [0]. I lectured mathematics for a while at a "former polytechnic" and used to enjoy leaving print-outs of this sort of paper in the faculty communal areas.

[0] https://www.axler.net/DwD.html

Note that "done right" means done with Axler's completely subjective and unusual hatred of determinants, chronicled here [0]. It is in no way "done right" in some definitive, rigorous way; most math professors I have spoken to either strongly disagree with the presentation or have no particular preference.

[0] https://www.axler.net/DwD.html

I got halfway though the exercises with the help of a reading group. They were very hard, bit thought provoking, so I would definitely recommend. Don't feel discouraged if you get stuck and try not to look at the solutions right away.

"Human verification failed" - so this is a broken link.

I found it really insightful (and always overlooked) to distinguish between vector and co-vector spaces. It doesn't necessarily produce new knowledge, but makes things more clear.

I wanted to learn the underlying principles of LLM/AI and got myself Shilov's book. Wow that was so thick, each paragraph took a while to figure. This could be a nice option..

Thanks!

Lately been deep diving into linear algebra. And a way which i engage with it is that I tell AI to generate interactive examples + questions on Lean or Haskell. Its so fun, just deriving the intuition in these languages.

I passed the class just because of how good the book is.

i literally threw this book in the trash cause it was too dense and pretentious.

This is a holy book for a lot of game developers.

Kindle format link == 404

Overrated and tendentious book. There are many better linear algebra texts. His polemic against determinants is poorly motivated, misguided, and distracting. The writing is quite formal and not terribly inspiring. The coverage is adequate but nothing more.

My bag of tricks is better than your bag of tricks. Alright.

As with most textbooks, it fails to motivate why reading it is worth the investment. Perhaps it is a millennial old tradition of the Greek mystery schools, that the rite of passage came by proving your commitment to material knowledge without anything but fate in the school itself as motivation.

Rigor before Worth.

(Yes this is a pet peeve of mine :)

These days, linear algebra done right should be accompanied with some CAS to view how algorithms are used.

Possibly paired with some numerical algebra free text (many on the Internet)

Do not get the latest edition, the layout and typesetting is atrocious!

Last night I was looking into what to read after or along with 3Blue1Brown's series of Linear algebra videos [1]

The contenders seems to be:

- Linear Algebra Done Right - Sheldon Axler

- Liner Algebra Done Wrong - Sergei Treil

- Introduction to Linea Algebra - Gilbert Strang

- Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares by Stephen Boyd and Lieven Vandenberghe

[1] https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

Previously on Hacker News:

Linear Algebra Done Right 58 points, July 2023, 4 comments https://news.ycombinator.com/item?id=36576114

Linear Algebra Done Right – 4th Edition, 631 points, Oct 2023, 294 comments https://news.ycombinator.com/item?id=38060159

Linear Algebra Done Right [pdf], 85 points, Sept 2024, 39 comments https://news.ycombinator.com/item?id=41416799

These days, linear algebra done right should be accompanied with some CAS to view how algorithms are used.

Possibly paired with some numerical algebra free text (many on the Internet)

I've taught out of the first three of these. If I had to pick one for a first study, I'd vote for Strang -- and watch his videos while you go. Our second-semester mostly-math-majors linear algebra course uses Axler, which I think is nice for the purpose, but our students already have done a semester of computational stuff first. (Though the complete absence of any computations in the book means they don't always connect the material from the two courses very well.)

As a math educator, I strongly dislike both Strang and Axler for a 1st course. I've heard great things about Strang's lectures, but his book is disorganized and too heavy on computation. Axler's book is wonderful, but as explicitly stated on the back cover, it's designed for a 2nd course and primarily aimed at math majors.

I recommend and teach my YouTube Live series out of Fraleigh [1], but unfortunately it's out of print. Lay seems to be a good modern alternative.

[1] https://linear.mathcanbeahobby.com

LADW saved me in undergrad, but I was pretty much exactly the target audience in an honors-level freshman math course:

"[per Treil, LADW is for] a student who, while not yet very familiar with abstract reasoning, is willing to study more rigorous mathematics than what is presented in a “cookbook style” calculus type course."

But yeah it's really attempting to introduce you to higher mathematics rather than get you comfortable doing linear algebra per se.

Strang is simpler and clearer. Axler is more advanced in the sense that it doesn’t tie it to matrices. Strange is a “first course” book, Axler is a second course.

Strang's lecture series are a nice and friendly accompaniment, especially if you don't have a reading group https://www.youtube.com/watch?v=7UJ4CFRGd-U&list=PL221E2BBF1...

LADW and LADR are great too, for an honors approach with more focus on proofs. To me it would make more sense on a second pass.

I really recommend Matrix Analysis and Applied Linear Algebra by Carl Meyer. It's both concise and comprehensive. Strange is very good, but feels kinda vague and long winded in comparison (very good for a high level understanding of the tools you're dealing with)

If you liked 3B1B and prefer intuition/applications-heavy view, then definitely Strang over Axler. Check out especially his newer textbook "Linear Algebra and Learning from Data".

Axler is more of a pure math textbook - if you want to dive more into proofs and abstractions.

I strong second Strang. His book is the best first introduction to linear algebra, with "Done Right" marketing itself as a second course. Axler is notoriously shy with matrices, but Axler introduces them up front and uses them for the rest of the book.

Lorenzo Sadun's Linear Algebra: The Decoupling Principle would probably be enough too if you added something about determinants.

Previously in my ~/Downloads:

linear_algebra_done_right.pdf, 0 pages read, July 2023

linear_algebra_done_right (1).pdf, 0 pages read, Oct 2023

linear_algebra_done_right (2).pdf, 0 pages read, Sept 2024

Downloading (3) now.

[0] https://www.axler.net/DwD.html

"Human verification failed" - so this is a broken link.

Determinants are easy to use but very hardly to grasp intuitively, this is not a minority point of view. See countless of StackOverflow questions begging for a conceptual exposition of determinants.

The easiest conceptual handle is geometric: volume expansion, but seeing how this is related to the sum over all permutations, or how those two point of views are related to a set of equations having a solution or not is not easy to see even in the 2D case.

I won't be surprised if math professors don't have this issue like you said (especially if someone is comfortable with wedge products), but the vast majority of newcomers who are interested in understanding why something works rather than just how to use it struggle all the time with determinants.

Yeah I feel like this "done right" part is responsible for most of the popularity of this book. Makes the reader think they've been learning it wrong. Kind of like these clickbait videos "you've been folding your laundry wrong your whole life!" or whatever

I found it really insightful (and always overlooked) to distinguish between vector and co-vector spaces. It doesn't necessarily produce new knowledge, but makes things more clear.

Kindle format link == 404

Do not get the latest edition, the layout and typesetting is atrocious!

i literally threw this book in the trash cause it was too dense and pretentious.

I passed the class just because of how good the book is.

This is a holy book for a lot of game developers.

My bag of tricks is better than your bag of tricks. Alright.

Rigor before Worth.

(Yes this is a pet peeve of mine :)

LADW saved me in undergrad, but I was pretty much exactly the target audience in an honors-level freshman math course:

But yeah it's really attempting to introduce you to higher mathematics rather than get you comfortable doing linear algebra per se.

Previously in my ~/Downloads:

linear_algebra_done_right.pdf, 0 pages read, July 2023

linear_algebra_done_right (1).pdf, 0 pages read, Oct 2023

linear_algebra_done_right (2).pdf, 0 pages read, Sept 2024

Downloading (3) now.

Lorenzo Sadun's Linear Algebra: The Decoupling Principle would probably be enough too if you added something about determinants.

Strang's lecture series are a nice and friendly accompaniment, especially if you don't have a reading group https://www.youtube.com/watch?v=7UJ4CFRGd-U&list=PL221E2BBF1...

LADW and LADR are great too, for an honors approach with more focus on proofs. To me it would make more sense on a second pass.

Yeah - otherwise it’s bizarre to have row vectors and column vectors.

My personal favorite is No Bullshit Guide to Linear Algebra - it gives a really good overview of math fundamentals and overall strikes a good balance between keeping things simple and giving enough insight to comfortably dig deeper in the topic.

I wanted to learn the underlying principles of LLM/AI and got myself Shilov's book. Wow that was so thick, each paragraph took a while to figure. This could be a nice option..

Thanks!

I recommend and teach my YouTube Live series out of Fraleigh [1], but unfortunately it's out of print. Lay seems to be a good modern alternative.

[1] https://linear.mathcanbeahobby.com

If you liked 3B1B and prefer intuition/applications-heavy view, then definitely Strang over Axler. Check out especially his newer textbook "Linear Algebra and Learning from Data".

Axler is more of a pure math textbook - if you want to dive more into proofs and abstractions.

Strang is simpler and clearer. Axler is more advanced in the sense that it doesn’t tie it to matrices. Strange is a “first course” book, Axler is a second course.

Shilov is a very popular recommendation, because it‘s good and very, very cheap (a Dover).

> His polemic against determinants is poorly motivated, misguided, and distracting.

What polemic? Defining the determinant as the unique multilinear alternating form satisfying certain properties is very normal (and in fact the only way that really makes sense for both finite- and infinite-dimensional vector spaces). There are zero unusual things with this book imo.

Something else you recommend?

Apparently, it's on Internet Archive: https://archive.org/details/linearalgebra0000fral/

agree, tried both and they are hard as a starter.

Is this the book you are referring to? https://epubs.siam.org/doi/book/10.1137/1.9781611977448. It has 991 pages, not exactly what I would call concise.

I would also say Axler is much better prep for higher level applied math, as well as pure. If you are interested in how the big ideas of linear algebra extend to things like Fourier analysis it's very helpful to see the more abstract explanation of vector spaces.

Depends how you think. I found Strang impenetrable and Axler simple and lucid. Some people seem to find abstract vector spaces weird and unmotivated without doing a load of stuff with lists and grids of numbers first. I find determinants weird and unmotivated without learning exterior algebra first. I wish Axler had been my first course.

Axler does limit itself to vector spaces over real and complex fields, though.

That‘s fine, but I would have appreciated notices, which proofs and theorems do not hold in the general case.

It‘s an exercise for the reader.

Determinants are easy to use but very hardly to grasp intuitively, this is not a minority point of view. See countless of StackOverflow questions begging for a conceptual exposition of determinants.

Or "this one simple trick makes Big Linear Algebra hate you"

Yeah - otherwise it’s bizarre to have row vectors and column vectors.

To me it doesn't seem wrong to call covectors "row vectors", but one shouldn't simply convert them to each other with no good reason.

If anyone is interested, check out the PDF preview here https://minireference.com/static/excerpts/noBSLA_v2_preview....

and the printable concept maps here: https://minireference.com/static/conceptmaps/linear_algebra_...

Shilov is a very popular recommendation, because it‘s good and very, very cheap (a Dover).

> His polemic against determinants is poorly motivated, misguided, and distracting.

Is this the book you are referring to? https://epubs.siam.org/doi/book/10.1137/1.9781611977448. It has 991 pages, not exactly what I would call concise.

Apparently, it's on Internet Archive: https://archive.org/details/linearalgebra0000fral/

Something else you recommend?

This polemic: Sheldon Axler -- Down With Determinants

https://www.axler.net/DwD.html

A strange and unpopular opinion.

I recommend getting the 1st edition. I read the 1st edition and loved it. Here is a relevant review of the 2nd edition https://janert.me/blog/2025/book-review-matrix-analysis-and-...

agree, tried both and they are hard as a starter.

Axler does limit itself to vector spaces over real and complex fields, though.

That‘s fine, but I would have appreciated notices, which proofs and theorems do not hold in the general case.

It‘s an exercise for the reader.

It may simply be that linear algebra is a difficult topic to become introduced to, and even the best teaching tools cannot make it easy.

Or "this one simple trick makes Big Linear Algebra hate you"

To me it doesn't seem wrong to call covectors "row vectors", but one shouldn't simply convert them to each other with no good reason.

If anyone is interested, check out the PDF preview here https://minireference.com/static/excerpts/noBSLA_v2_preview....

and the printable concept maps here: https://minireference.com/static/conceptmaps/linear_algebra_...

Nice seeing you on HN. I thoroughly enjoyed the No Bullshit LA and Maths & Physics books. I wish more books would include concept maps, I find them a very useful tool for understanding how things fit in the broader picture.

I recommend getting the 1st edition. I read the 1st edition and loved it. Here is a relevant review of the 2nd edition https://janert.me/blog/2025/book-review-matrix-analysis-and-...

This polemic: Sheldon Axler -- Down With Determinants

https://www.axler.net/DwD.html

A strange and unpopular opinion.

It may simply be that linear algebra is a difficult topic to become introduced to, and even the best teaching tools cannot make it easy.

Yes, concept maps are the bee's knees.

I remember from my tutoring days how useful they were to organize the different concepts covered in each lesson: I would start with a blank sheet and make the student add concepts to it as the lesson progressed, then by the end of the lesson use the concept map to review what we learned. Specifically, I would ask them to explain in their own words each "arrow" which was a great way to uncover misconceptions and solidify the material.

Once I'm done with editing the current book[1], I hope to have time to work on making dynamic concept maps that you can click on and explore/zoom-in on. I feel it would be cool to jump between detailed view (concepts), intermediate scale (topics), and high-level view (subjects).

[1] https://noBSstats.com

Yes, concept maps are the bee's knees.

[1] https://noBSstats.com

Sheldon Axler

The fourth and most recent edition of Linear Algebra Done Right is an Open Access book available in English, Chinese, Farsi, Greek, and Portuguese. Other languages are coming soon.

Electronic versions of this fourth edition, which has a Creative Commons BY-NC license, are legally availble without cost at the links below.

English: PDF file for Linear Algebra Done Right, fourth edition (free; 16 August 2026)
English: Kindle version of Linear Algebra Done Right, fourth edition (free; 2024)
Chinese: PDF file for Linear Algebra Done Right, fourth edition, in Chinese; translated by Oliver Wu and Yang He (free; 1 June 2026)
Farsi: PDF file for Linear Algebra Done Right, fourth edition, in Farsi; translated by Alireza Takrimi, Rabert Khamounejad, and Illatra Khamounejad (free; 20 June 2026)

The fourth edition of Linear Algebra Done Right contains over 250 new exercises and over 70 new examples, along with several new topics and multiple improvements throughout the book. See page xvi in the English file linked above for a list of major improvements and additions in the fourth edition.

The hardcover print version of the fourth edition of Linear Algebra Done Right is available from Amazon at the link below.

English: print version of Linear Algebra Done Right, fourth edition

Print versions of translations of the fourth edition into Greek and into Portuguese are available at booksellers in appropriate countries.

Print versions of translations of the third edition into Basque and Chinese are available from Amazon at the links below.

This best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra.

No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner product spaces are then introduced, leading to the finite-dimensional spectral theorem and its consequences such as the singular value decomposition. Generalized eigenvectors are then used to provide insight into the structure of a linear operator. Determinants are cleanly introduced via alternating multilinear forms.

Excerpts from Reviews

Altogether, the text is a didactic masterpiece.
zbMATH

Axler demotes determinants (usually quite a central technique in the finite dimensional setting, though marginal in infinite dimensions) to a minor role. To so consistently do without determinants constitutes a tour de force in the service of simplicity and clarity; these are also well served by the general precision of Axler's prose... The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library.
Choice

The determinant-free proofs are elegant and intuitive.
American Mathematical Monthly

Clarity through examples is emphasized... the text is ideal for class exercises... I congratulate the author and the publisher for a well-produced textbook on linear algebra.
Mathematical Reviews

If you liked the previous editions, you will like this new edition even better!
Monatshefte für Mathematik

customer reviews from Amazon

About the Author
textbook adoptions: list of 430 universities and colleges that have used Linear Algebra Done Right as a textbook
videos to accompany the book
errata

Follow @AxlerLinear on X (Twitter) for updates on this book.

Linear Algebra Done Right usually has the best Amazon sales rank of any linear algebra book at this level. The hardcover version of Linear Algebra Done Right usually costs considerably less than hardcover versions of competing linear algebra textbooks.

This book is partly based on ideas contained in my paper Down with Determinants!, published in the American Mathematical Monthly. This paper received the Lester R. Ford Award for expository writing from the Mathematical Association of America.

Questions or comments about Linear Algebra Done Right can be sent to the author at [linear@axler.net](mailto:linear@axler.net).

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Linear algebra done right

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Sheldon Axler

Excerpts from Reviews