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.
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.)
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.
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.
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 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.
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.
The nature of textbooks is that each one is better suited for a certain profile of reader. It depends a lot on the way the reader has learnt to learn things until that point in their life.
If you liked 3B1B's style, you will prefer strang over axler. Axler and treil to a greater extent focus on bringing out the abstract elegance and the kind of rigour a math major enjoys. Strang's book also has videos accompanying - on MIT OCW.
B&V VMLS on your list is interesting - they focus a lot on real-world instantiations of the concepts and have you code up things in the (excellent) exercises. Depending on your goals, you can do only this, or strang and then this. Definitely look at the exercises in any case though.
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.
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
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 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.
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.
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 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..
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.
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.
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.
> 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.
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.
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...
I found Linear Algebra by Friedberg, Insel and Spence to be excellent. Very clear, modern notation, great exercises. It's also what Tao lectured from in 115A: https://www.math.ucla.edu/~tao/resource/general/115a.3.02f/
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
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.)
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.
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.
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)
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.
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 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.
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.
+1 for Strang.
Lorenzo Sadun's Linear Algebra: The Decoupling Principle would probably be enough too if you added something about determinants.
You zoomers are making a list of linear algebra books and not citing Lang? Get off my lawn ;)
Would you mind sharing your thoughts more?
The nature of textbooks is that each one is better suited for a certain profile of reader. It depends a lot on the way the reader has learnt to learn things until that point in their life.
If you liked 3B1B's style, you will prefer strang over axler. Axler and treil to a greater extent focus on bringing out the abstract elegance and the kind of rigour a math major enjoys. Strang's book also has videos accompanying - on MIT OCW.
B&V VMLS on your list is interesting - they focus a lot on real-world instantiations of the concepts and have you code up things in the (excellent) exercises. Depending on your goals, you can do only this, or strang and then this. Definitely look at the exercises in any case though.
+1 for Boyd
Previously on Hacker News:
Linear Algebra Done Right 58 points, July 2023, 4 comments https://news.ycombinator.com/item?id=36576114
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Linear Algebra Done Right [pdf], 85 points, Sept 2024, 39 comments https://news.ycombinator.com/item?id=41416799
Previously in my ~/Downloads:
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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
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
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.
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_...
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.
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
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
Or "this one simple trick makes Big Linear Algebra hate you"
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 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.
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.
"Human verification failed" - so this is a broken link.
This is a holy book for a lot of game developers.
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.
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.
> 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.
This polemic: Sheldon Axler -- Down With Determinants
https://www.axler.net/DwD.html
A strange and unpopular opinion.
Something else you recommend?
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 literally threw this book in the trash cause it was too dense and pretentious.
Kindle format link == 404
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 :)
Do not get the latest edition, the layout and typesetting is atrocious!