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Linear Algebra Done Right – 4th Edition

631 points by __rito__ 3y ago

threatofrain 3y ago
Note that Axler intended this book to be the second reading of Linear Algebra after you've already taken a first course, but it is doable for a first reading. If you want to be crazy you can also check out A (Terse) Introduction to Linear Algebra by Katznelson & Katznelson.
- uoaei 3y ago
  Dang, I remember everyone not enjoying linear algebra class with Katznelson in undergrad. I did ok but it felt like way more focus on things like row elimination algorithms than why any of it works. It wasn't until I worked with a PhD geometer that any of it made sense and they largely cribbed from Linear Algebra Done Right. Hopefully the book is better than a class aimed at a generic mix of STEM undergrads.
  - deleted 3y ago
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- tptacek 3y ago
  I did undergrad linear algebra with my daughter last semester, and Strang and Axler were a good one-two punch, Strang for the computation, Axler for the proofs homework.
  - basedbertram 3y ago
    So did you read through Strang and then read Axler, or did you try to work through both of them at the same time?
    - tptacek 3y ago
      Same time. I'd taught myself linear algebra from the Strang lectures (and a Slack study group we set up with some random university's syllabus, which gave us a set of homework problems to do) long before this, so mostly I just matched the professor's lectures to the Strang material, and dipped in and out of Axler when proof and conceptual stuff came up; it's not like we did Axler cover to cover. Before doing this, I'd only ever sort of skimmed Axler; it's sort of not the linear algebra you care about for cryptography, and up until the spectral theorem stuff that's exactly what Strang was. It was neat to get an appreciation for Axler this was.
  - qbit42 3y ago
    Yeah, my math class followed Axler, which was great - but I didn't really get a feel for how useful linear algebra was until I read through Strang on my own. The applications are endless!
    - naijaboiler 3y ago
      >>The applications are endless! Pretty much all of grad level engineering classes, all of grad level Stats classes, all of grad level Econ classes include applications of LA. Heck quantitative research work in social sciences are still included via stats.
  - selimthegrim 3y ago
    When I taught linear algebra I used Strang and a book from the UK by Morris that seems to be out of print these days.
- tripdout 3y ago
  Wow, we used this textbook (albeit alongside a more beginner focused one) in first year undergrad Computer Science.
- anta40 3y ago
  What's your recommendation material for first course? Strang, perhaps? I used Howard Anton's.
  - mayd 3y ago
    I would definitely choose Strang to begin learning Linear Algebra. He develops the geometric intuition you need for applying it in other subjects such as Calculus or Statistics or Physics. Linear Algebra is not just Abstract Algebra, at least for most of us, including most mathematicians. After Strang, or even alongside him, if you want more rigour then you would benefit from a text like that of Anton, or Friedberg, or Curtis or, God forbid, Axler. FYI I bought Axler's book (3rd edition).
febra_ 3y ago
I have this book. It's been more than great. I fully recommend it to anyone looking for a quick and simple introduction to linear algebra
some_math_guy 3y ago
Like basically everybody else I teach out of this book, and I'm happy to see a new edition. I'm curious what's changed/added -- I already am unable to get through the whole thing in a semester. At our school students take a computational linear algebra course first (with a lot of row reduction). So I am slowed down a bit by constantly trying to help the students see that the material is really the same thing both times through. I do wish there were a little more of that in Axler.
- blovescoffee 3y ago
  I've studied from this book. Since you're teaching out of it, I'm curious if you've read/have an opinion on Strang's books. I love his lectures :)
  - some_math_guy 3y ago
    Sure, I am very familiar with them -- I actually TAed 18.06 for Strang once upon a time. They're great books too. Which is better is mostly a question of what point of view you're after -- if you want to actually calculate anything, Axler's book is not going to help you, but if you want a more conceptual view of the subject it's best place. If you're really serious about learning linear algebra, you probably want to read both, first Strang, then Axler.
    - blovescoffee 3y ago
      That's awesome. I bet TAing for Strang was a great experience. Now I'm curious if you have any recommendations for "actually calculating something"
seo-speedwagon 3y ago
The 3rd edition of this book is what my undergraduate linear algebra course used. It was a fantastic book. I feel like more computation- and determinant-heavy approaches can make the subject feel like a slog, but this book made me really enjoy, appreciate, and get a gut-level understanding of the subject
gmiller123456 3y ago
It's not at all obvious from the headline, but the news is that the book is FREE, you can download the PDF from the first link.
cyberax 3y ago
I skimmed it, and it's a bit too succinct for my liking. I guess it's inevitable if you want to try to pack that much material into one book.
__rito__ 3y ago
This book is Open Access and you can download it from this link [0]. [0]: https://link.springer.com/content/pdf/10.1007/978-3-031-41026-0.pdf https://link.springer.com/content/pdf/10.1007/978-3-031-4102...
- Alifatisk 3y ago
  Amazing!
sieste 3y ago
The look on Christina of Sweden's face on page 1 made me laugh, she looks exactly how you would imagine a 17th century princess hearing about Linear Algebra.
nextos 3y ago
I dislike the typesetting changes that came after the second edition. It was a really elegant book, reminiscent of other classic Springer Undergraduate Mathematics Series tomes. Lots of distracting color, highlighting, and boxes were added, which IMHO make the book less clear. Obviously, the content is still great.
- fiforpg 3y ago
  Yep. Huge kudos to the author for making it available, but the PDF does feel sloppy with all the bright colors and images. In science textbooks, less is more. Compare this to the very latest edition of Stewart's calculus, which now uses even more pastel, subdued colors for diagrams.
  - nextos 3y ago
    Exactly, a calm black and white design does not need to be unfriendly. Hubbard & Hubbard or MacKay are two examples of beginner-friendly books with great typesetting.
- blt 3y ago
  with the power of `tcolorbox` comes great responsibility.
readthenotes1 3y ago
Linear Algebra Done Right This Time We Promise (4th try) J/k ;)
- selimthegrim 3y ago
  4th sequence
navanchauhan 3y ago
For the course “Linear Algebra for Math Major”, we are using both LADR and Linear Algebra Done Wrong (Open Access as well)
mike986 3y ago
I have surveyed every LA books out there and a lot of amazons reviews claimed axler’s book is the best LA book. It might be for case for printed books for sale. But I stumbled upon Terrance Tao’s pdf LA lecture slides on his website and it is so much better than all the books I’ve surveyed. The writing is super clear and everything is built from the first principles. (BTW terry’s real analysis book did the same for me. Much more clear and easy to follow than the classics out there)
- abdullahkhalids 3y ago
  I believe these are notes that you are referring to https://terrytao.files.wordpress.com/2016/12/linear-algebra-notes.pdf https://terrytao.files.wordpress.com/2016/12/linear-algebra-...
  - pstuart 3y ago
    Thanks!
  - havercosine 3y ago
    These notes are excellent. One good thing is how often Terence Tao gives real life examples and analogies, contrary to what one may expect from a fields medal winner. From utilitarian perspective, reading Axler's book looks like comically bad use of one's time.
- zeroonetwothree 3y ago
  I'm not sure that Axler's book is great as a first LA book. I would go with something more traditional like Strang. Although I really didn't feel like I "got" LA until I learned algebra (via Artin). By itself LA feels very "cookbook-y", like just a random set of unrelated things. Whereas in the context of algebra it really makes a lot more sense.
  - mohamez 3y ago
    >I'm not sure that Axler's book is great as a first LA book. Linear Algebra Done Right is a text for beginners who want to study linear algebra in a proof based, mathematically rigorous way. So, if you want that I think it's a good fit as a first linear algebra book.
fiforpg 3y ago
About avoiding determinants to the degree that this book does: while I agree it makes sense to delay introducing them, the goal should not be avoidance but clarity. The way author has to bend himself backwards here when dealing with eigenvalues isn't great either. I would recommend Strang for a healthy balance in handling determinants.
- jjoonathan 3y ago
  I read Strang and then Axler. Strang is great at numerics but weak at presenting the abstract picture. I feel like if I had taken, say, finite elements (or any other subject where it's important to take the abstract / infinite dimensional picture seriously before reducing to finite dimensions) right after Strang without reading LADR then I'd have been seriously underprepared.
  - fiforpg 3y ago
    You have a point in that to understand any particular subject well, it makes sense to read more than one book on it, at least to compare the different perspectives. Also worth noting that Strang has a couple of similar linear algebra books, so we might not even be discussing the same text.
    - jjoonathan 3y ago
      That's entirely possible, but in the context of introductory books I think it's fair to assume & limit scope to Strang's "Introduction to Linear Algebra" and Axler's "Linear Algebra Done Right." I am an applications-oriented person and my inclination was to go directly from a matrix/determinant heavy picture into applications. Strang['s intro text] only. I am extremely glad that someone intercepted me and made me get some practice with abstract vector spaces, operators, and inner product spaces first, using Axler. This practice bailed me out and differentiated me from peers on a number of occasions, so I want to pass down the recommendation.
  - nerdponx 3y ago
    FWIW I think this is the benefit of Strang. If you're in science or engineering or statistics, often you don't need the general picture, and IMO too much generality gets in the way of understanding. Start with a good understanding of the most important cases that appear in applied work, and drill them until you're fluent with them. Then generalizing will be easier.
imjonse 3y ago
From the preface. "You cannot read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast." Sadly, he's probably right.
- zeroonetwothree 3y ago
  Meh, there's different goals you could have. I actually find it enjoyable to read math more quickly (almost like a novel) which gives you a good sense of a lot of the higher-level themes and ideas. Then if it's interesting I might spend more time on it.
  - mohamez 3y ago
    He is talking about reading it after you decided that it is interesting.
- mohamez 3y ago
  Tips on Reading Mathematics[1]: - Be an active reader. Open to the page you need to read, get out some paper and a pencil. - If notation is defined, make sure you know what it means. Your pencil and paper should come in handy here. - Look up the definitions of all words that you do not understand. - Read the statement of the theorem, corollary, lemma, or example. Can you work through the details of the proof by yourself? Try. Even if it feels like you are making no progress, you are gaining a better understanding of what you need to do. - Once you truly understand the statement of what is to be proven, you may still have trouble reading the proof—even someone’s well-written, clear, concise proof. Try to get the overall idea of what the author is doing, and then try (again) to prove it yourself. - If a theorem is quoted in a proof and you don’t know what it is, look it up. Check that the hypotheses apply, and that the conclusion is what the author claims it is. - Don’t expect to go quickly. You need to get the overall idea as well as the details. This takes time. - If you are reading a fairly long proof, try doing it in bits. - If you can’t figure out what the author is doing, try to (if appropriate) choose a more specific case and work through the argument for that specific case. - Draw a picture, if appropriate. - If you really can’t get it, do what comes naturally—put the book down and come back to it later. - You might want to take this time to read similar proofs or some examples. - After reading a theorem, see if you can restate it. Make sure you know what the theorem says, what it applies to, and what it does not apply to. - After you read the proof, try to outline the technique and main idea the author used. Try to explain it to a willing listener. If you can’t do this without looking back at the proof, you probably didn’t fully understand the proof. Read it again. - Can you prove anything else using a similar proof? Does the proof remind you of something else? - - What are the limits of this proof? This theorem? - If your teacher is following a book, read over the proofs before you go to class. You’ll be glad you did. [1] Reading, Writing, and Proving: A Closer Look at Mathematics By Ulrich Daepp and Pamela Gorkin.
photochemsyn 3y ago
This is linear algebra for undergraduate math majors, but if you just want an basic understanding of the topic with a focus on computational applications, Poole's "Linear Algebra: A Modern Introduction" is probably more suitable as it's heavy on applications, such as Markov chains, error-correcting codes, spatiel orientation in robotics, GPS calculations, etc. https://www.physicsforums.com/threads/linear-algebra-a-modern-introduction-by-david-poole.665360/ https://www.physicsforums.com/threads/linear-algebra-a-moder...
HybridCurve 3y ago
If you think a textbook is good, the digital edition should available for review as they will often be invaluable in hardcopy for continued reference. I am pleased the author adheres in some part to this mode of thinking.
Buttons840 3y ago
I've been waiting for this for 6th months. Thanks to Sheldon Axler for making it available for free. This is intended to be a second book on Linear Algebra. For a first book I suggest "Linear Algebra: Theory, Intuition, Code" by Mike X Cohen. It's a bit different than a typical math textbook, it has more focus on conversational explanations using words, although the book does have plenty of proofs as well. The book also has a lot of code examples, which I didn't do, but I did appreciate the discussions related to computing; for example, the book explains that several calculations that can be done by hand are numerically unstable when done on computers (those darn floats are tricky). For the HN crowd, this is the right focus, math for the sake of computing, rather than math for the sake of math. One insight I gained from the book was the 4 different perspectives of matrix multiplication. I had never encountered this, not even in the oft-suggested "Essence of Linear Algebra" YouTube series. Everything I had seen explained only one of the 4 views, and then I'd encounter a calculation that was better understood by another view and would be confused. It still bends my mind to think all these different perspectives describe the same calculation, they're just different ways of interpreting it. At the risk of spamming a bit, I'll put my notes here, because this is something I've never seen written down elsewhere. The book has more explanation, these are just my condensed notes. 4 perspectives on matrix multiplication ======================================= 1 Element perspective (all possible dot / inner products) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (row count × row length) × (column length × column count) In AB, every element is the dot product of the corresponding row of A and column of B. The rows in A are the same length as the columns in B and thus have dot products. 2 Layer perspective (sum of outer product layers) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (column length × column count) × (row count × row length) AB is the sum of every outer product between the corresponding columns in A and rows in B. The column count in A is the same as the row count in B, thus the columns and rows pair up exactly for the outer product operation. The outer product does not require vectors to be the same length. 3 Column perspective (weighted sums / linear combinations) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (column length × column count) × (column length × column count) In AB, every column is a weighted sum of the columns in A; the weights come from the columns in B. The weight count in the columns of B must match the column count in A. 4 Row perspective (weighted sums / linear combinations) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (row count × row length) × (row count × row length) In AB, every row is a weighted sum of the rows in B; the weights come from the rows in A. The weight count in the rows of A must match the row count in B.
mohamez 3y ago
Linear Algebra Done Right is a good book for people who want to study the subject of linear algebra in a proof based, mathematically rigorous way. Here [1] you can find Sheldon Axler himself explaining the topics of the book in his YouTube channel! How wonderful is that! Here [2] you can find the solutions to the exercises in the book. [1] https://www.youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmSinFV0wEir2Vw https://www.youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmS... [2] http://linearalgebras.com/ http://linearalgebras.com/
- harry8 3y ago
  > Here [2] you can find the solutions to the exercises in the book. > >[2] http://linearalgebras.com/ http://linearalgebras.com/ I have trouble with believing an author has the students' interests at heart when the answers to solutions are not /in/ the book. Why on earth not? It makes no sense from any student perspective. I know it's common in textbooks to not even have them available at all. Any ideas why these are the norm not putting them where they clearly belong? It just seems so hostile.
  - layer8 3y ago
    It’s too easy to just look up the solution instead of being forced to think hard about the problem. This is training for when you’ll later encounter problems outside of textbooks where you’ll have no choice but to solve them on your own. And you’re supposed to have teaching assistants or similar available when you really remain stuck.
    - harry8 3y ago
      Yeah sorry that's just total and utter BS. No sale. I am an adult. >99% of students of university subjects are adults. Even assuming your premise totally, anyone so incredibly stupid they can't have it explained in the text when is the right time to use the solutions is too stupid to learn from the text. Axler puts them on a website that is less than 30 seconds away. "No choice..." Describes what proportion of texbook sales?
ayakang31415 3y ago
This is a great book, but you must first be familiar with proof-focused mathematics (logic and set theory). If you are not, I first suggest you study a book called "How to Prove It: A Structured Approach" by Daniel J. Velleman before studying LA Done Right.
- noqc 3y ago
  You need a setting in which to learn proof based mathematics, and linear algebra really is the first place where students are ready for that journey. Not everyone is going to be able to do it, but it's very incorrect to say that you must be familiar with proof. One must start somewhere, and ZF ain't it.
  - ayakang31415 3y ago
    Sorry, what is ZF?
    - noqc 3y ago
      Zermelo-Fraenkel, the axioms of set theory (minus choice). This would be what you were teaching if you were to start teaching mathematics by teaching its (classical) foundations.
      - ayakang31415 3y ago
        I briefly checked the axioms, and if I recall correctly, Velleman's book does not delve into ZF in detail, but it is a beginner friendly book to give enough exposure to quantitative logic and set theory to feel comfortable working on proof-based mathematics such as LA (not survey version of LA, which should be called matrix algebra instead).
        noqc 3y ago
        Sorry, I wasn't talking about Velleman's book on whatever, I was just challenging the sentiment that you should learn the "basics of proof based math" before linear algebra. Linear algebra is the basics of proof based math. Edit: I should clarify that I believe that there are many approaches to learning mathematics, I'm sure that Velleman's book is fine, just that it isn't true that you should not try to learn from Axler, although perhaps someone who has a bit less of an axe to grind.
ekm2 3y ago
This is good as a second course on Linear Algebra.For a first course,use (I am not kidding) Linear Algebra Done Wrong by Sergei Treil https://www.math.brown.edu/streil/papers/LADW/LADW.html https://www.math.brown.edu/streil/papers/LADW/LADW.html
- endymi0n 3y ago
  omg just had a look and this one is just everything I hate about mathematics and academia. Starts with lots of random definitions, remarks, axioms and introducing new sign language while completely disregarding introducing what it‘s supposed to do, explain or help with. All self-aggrandization by creating complexity, zero intuition and simplification. Isn‘t there anybody close to the Feynman of Linear Algebra?
  - ekm2 3y ago
    There is no royal road bro
    - 77pt77 3y ago
      In his mind his above royals, so a royal road would be demeaning to him.
  - PartiallyTyped 3y ago
    What about Axler’s then?
  - gadrev 3y ago
    Gilbert Strang's course on Linear Algebra. Playlist: https://www.youtube.com/playlist?list=PL49CF3715CB9EF31D https://www.youtube.com/playlist?list=PL49CF3715CB9EF31D Not as big in scope, though, but great introduction.
  - bscphil 3y ago
    Yeah, a good example is on the second page of the first chapter: > Remark. It is easy to prove that zero vector 0 is unique, and that given v ∈ V its additive inverse −v is also unique. The is the first time the word "unique" is used in the text. Students are going to have no idea whether this is meant in some technical sense or just conventional English. One can imagine various meanings, but that doesn't substitute for real understanding. This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. On the surface the opposite is true - you complain, for instance, that the text jumps immediately into using technical language without any prior introduction or intuition building. My take is that intuition building doesn't need to replace or preface the use of formal precision, but that what is needed is to bridge concepts the student already understands and has intuition for to the new concept that the student is to learn. In terms of intuition building, I think it's probably best to introduce vectors via talking about Euclidean space - which gives the student the possibility of using their physical intuitions. The student should build intuition for how and why vector space "axioms" hold by learning that fundamental operations like addition (which they already grasp) are being extended to vectors in Euclidean space. They already instinctively understand the axiomatic properties being introduced, it's just that the raw technical language being thrown at them fails to connect to any concept they already possess.
tenderfault 3y ago
Right. “Complex numbers were invented so that we can take square roots of negative numbers” Why? I wish a textbook tell me why, right there.
- jjoonathan 3y ago
  Because complex numbers make the fundamental theorem of Algebra nice and simple rather than complicated and ugly. In turn, this makes the spectral theorem of linear algebra nice and simple rather than complicated and ugly. In turn, this makes a bunch of downstream applications nice and simple rather than complicated and ugly. You will get a feel for this if you work Axler's problems. More importantly, you will gain an intuition for the fact that if you turn up your nose at complex numbers while going into these application spaces, you are likely to painstakingly reinvent them except harder, more ugly, and worse. Example: in physics, oscillation and waves A. underpin everything and B. involve energy sloshing between two buckets. Kinetic and potential. Electric and magnetic. Pressure and velocity. These become real and imaginary (or imaginary and real, it's arbitrary). This is where complex numbers -- where you have two choices of units -- absolutely shine. Where you would have needed two coupled equations with lots of sin(), cos(), trig identities, and perhaps even bifurcated domains you now have one simple equation with exponentials and lots of mathematical power tools immediately available. Complex numbers are a huge upgrade, and that's why anything to do with waves will have them absolutely everywhere.
  - wwarner 3y ago
    You might think that in every real world application, complex numbers are introduced as a convenience, and that every calculation that takes advantage of them ends with taking the real part of the result, but that's not the case. In QM, the final answer contains an imaginary part that cannot be removed.
    - BeetleB 3y ago
      Having studied QM, I disagree. You can always introduce trigonometry and remove it. Keep in mind that everything observable is a real number even if the intermediate calculations involve imaginary numbers.
MrBlueIncognito 3y ago
I don’t know if it’s just me, but I’m terribly lost in the search for the right texts. Every time I come across a new book/resource, it compounds the confusion. I find myself incapable of sitting down with a book and working through it without switching to another book in-between. I’d be really happy to hear if anyone has found a solution to this unproductive but sticky habit.
generationP 3y ago
4th edition and still hasn't bothered to disambiguate between polynomial and polynomial function. Still a good example of mathematical writing. But, as Einstein supposedly said, "as simple as possible but not simpler". Why is it always American authors that forget that last part?
- beyonddream 3y ago
  Do you suggest any book that does disambiguate between polynomial and polynomial function (i thought they were equivalent!) ?
  - adrian_b 3y ago
    I do not know what the previous poster has meant to say, but polynomials can be defined in a more abstract way as just elements of certain vector spaces on which an additional operation of multiplication between polynomials is defined, which obeys special rules (the "carry-less multiplication", which is implemented in hardware by most modern CPUs, is an example of polynomial multiplication). In computers there are many applications of such abstract polynomials, e.g. for error detecting or correcting codes, pseudo-random number generation, authenticated encryption and others, which depend only on the rules for addition, internal multiplication and multiplication with scalars, and which have nothing to do with the polynomial functions associated with the polynomials.
slowhadoken 3y ago
the good old indispensable and underappreciated linear algebra
NoNameHaveI 3y ago
To the author: I commend you on your work! As a former full-time educator, I believe that one of the best thing you can do to promote usage is to create supplementary materials (lecture slides, quiz deck in Blackboard format, etc. I do realize there are differences between everyone, but when I was an educator, there were many times when I was expected to teach multiple subjects that I had little to no experience with, with as little as one days notice in many cases (the adjunct had a baby. Can you teach this?). TLDR; Having a good text is a good start. To increase adoption and usage, create teaching materials, which saves valuable time for instructors. Peace out
nixer6872 3y ago
Just read Dummit Foote. L.A. Done Right is not that good as its fancy title.
arnonejoe 3y ago
The linear algebra refresher course on Udacity is excellent (and free). https://www.udacity.com/course/linear-algebra-refresher-course--ud953 https://www.udacity.com/course/linear-algebra-refresher-cour...
deleted 3y ago
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magnio 3y ago
A lot of the discordance in this thread can be boil down to "different people having different goals when reading". As for Linear Algebra, due to its wide relevance, it is hard for one book to accommodate to all needs of people from different fields and levels of mathematical maturity. I would say there are 2 main reasons for one to study LA: - To understand and use its toolbox in practice: in fields like computer graphics, engineering, statistics, data analysis, modeling, etc. - To understand and use its toolbox in other fields of (pure) math: single, multivariable, and functional analysis, abstract algebra, etc. (Sometimes the latter goal is coupled with a secondary goal of introducing the student to mathematical proof. However, for the self-learners, I don't think using a LA book is good for learning how to write proof for the first time.) Unlike in calculus where Spivak serves as the one-size-fits-all, readers should identify their reasons before picking an LA book. I have not much experience with the first goal. Strang's book seems quite popular, and given the title, I would say it fits this goal. Axler's book falls squarely into the second goal, and I wager it's a good book for this purpose if not for two things: his avoidance of determinants and the dearth of any computational aspects or applications (yes, even theoretical LA books ought to be grounded in reality). There are three books that, in my opinion, do better than Axler here: - "Linear Algebra" by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence: dry, does not instill in me the sense of excitement that Axler does. However, it is comprehensive and rigorous, with a decent amount of explanation and a good selection of exercises. Application aspects are present but quite disparate and incoherent. This is the old and long-standing recommendation. - "Linear Algebra" by Elizabeth S. Meckes and Mark W. Meckes: a new book that somehow manages to combine the material of Friedberg and the eloquence of Axler. The rigor and exercises are comparable to Friedberg, but the applications are presented much better here. - "Linear Algebra" by Sterling K. Berberian: a little bit more abstract than the other two, but the exposition is still very clear. It connects LA to other fields of math and gives plenty of examples and motivations (shocking I know). The exercises department is good. Honorable mentions to the classics of Serge Lang and Paul Halmos. They are concise and, as with the rest of their works, wonderfully written. For "Linear Algebra Done Wrong", well, I feel like it's a stripped down Friedberg: shorter but not better. (The proof of the multiplicativity of determinant is just some hand-waving at "a lucky coincidence".)
bigbacaloa 3y ago
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akasakahakada 3y ago
Seriously this is the worst book ever in the textbook history. People always laugh for stuff like "tensor is stuff that works like a tensor", but this book is literally doing that in every page.
bsaul 3y ago
Any advice for calculus ? something that would take you from derivatives up to the kind of advanced analysis required in quantum field theory ?
lysecret 3y ago
In my opinion the best course on LA by far: https://youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS14F4NxAZudSmv&si=qDiayasMQw7P-sJ_ https://youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS14F4N...
Ste_Tokyo 3y ago
The book starts out writing vectors as row vectors (1 x n matrices), then later introduces the matrix of a vector as a column vector. Just represent vectors as column vectors from the start, define R^n as R^n,1 (in the notation of the book) and then there's no confusion and you get matrix multiplication of column vectors as students learned in high school: M.x where M is an m x n matrix and x is an n x 1 column vector, with product an m x 1 column vector. Stick to this all the way through a first linear algebra textbook and undergrad tears can be avoided. e.g. example 3.32 in the text: T(1,0) = (1,2,7). Just write these as column vectors and the matrix multiplication is easier to visualize. (Also, T acting on a real number was written T(x) with brackets so T acting on a vector should really be written T((1,0)). Inconsistencies in notation like this lead to pain.) Then, for example, when you want to show that the rows of a matrix are independent, you can use a row vector to write succinctly x.M = 0 => x = 0 where x is now a row vector. And it's clear we're doing something other than transforming space vectors here as we've used column vectors for vectors representing points in space. For undergrad textbooks you want to decide on a convention, one that matches what was done in high school preferably, and stick to it.
- uxp8u61q 3y ago
  You do realize it makes no difference whether the vectors are written horizontally or vertically? Part of mathematical maturity is not getting hung up on pointless notation like this.
  - Ste_Tokyo 3y ago
    Clearly the goal of a first course in linear algebra is not mathematical maturity. If you don't set up the notation consistently from the start, once you start introducing change of bases formulae or proving row rank = column rank, etc., some students will get confused. See all the questions from confused students on Maths Stack Exchange about this point. The way this particular author has handled the issue is perverse: first he writes vectors as row vectors, then he introduces a mapping M(v) which maps the vector v to its representation as a column vector. Also, as I tangentially noted, he misses out the brackets for a function writing T(0,1) instead of T((0,1)), which is just incompetent. The book should be titled "Algebra Done Shite" not "Algebra Done Right" ;)
nimish 3y ago
The elusive linear algebra book that isn't simply matrix algebra. A gem.
ivzhh 3y ago
This book and mathematical analysis from Zorich are my favorite two math books.
RedCinnabar 3y ago
I know this book is not intended to be used as a first reading on Linear Algebra but, to me, this text isn’t good even for a second(or a third) read. The author hurries up too much on certain parts. I think that Serge Lang’s Linear Algebra does a better job in explaining pretty much every topic of the subject.
allangrant 3y ago
I. X
anjanb 3y ago
There's so much discussion here but I couldn't decide what is a good first course/book on linear algebra. Can someone please summarize and share a couple of books/courses ? Thanks
fattah25 3y ago
Everything good if it's free.

631 points by __rito__ 3y ago

Note that Axler intended this book to be the second reading of Linear Algebra after you've already taken a first course, but it is doable for a first reading. If you want to be crazy you can also check out A (Terse) Introduction to Linear Algebra by Katznelson & Katznelson.

uoaei 3y ago
Dang, I remember everyone not enjoying linear algebra class with Katznelson in undergrad. I did ok but it felt like way more focus on things like row elimination algorithms than why any of it works. It wasn't until I worked with a PhD geometer that any of it made sense and they largely cribbed from Linear Algebra Done Right. Hopefully the book is better than a class aimed at a generic mix of STEM undergrads.
- deleted 3y ago
  [deleted]
tptacek 3y ago
I did undergrad linear algebra with my daughter last semester, and Strang and Axler were a good one-two punch, Strang for the computation, Axler for the proofs homework.
- basedbertram 3y ago
  So did you read through Strang and then read Axler, or did you try to work through both of them at the same time?
  - tptacek 3y ago
    Same time. I'd taught myself linear algebra from the Strang lectures (and a Slack study group we set up with some random university's syllabus, which gave us a set of homework problems to do) long before this, so mostly I just matched the professor's lectures to the Strang material, and dipped in and out of Axler when proof and conceptual stuff came up; it's not like we did Axler cover to cover. Before doing this, I'd only ever sort of skimmed Axler; it's sort of not the linear algebra you care about for cryptography, and up until the spectral theorem stuff that's exactly what Strang was. It was neat to get an appreciation for Axler this was.
- qbit42 3y ago
  Yeah, my math class followed Axler, which was great - but I didn't really get a feel for how useful linear algebra was until I read through Strang on my own. The applications are endless!
  - naijaboiler 3y ago
    >>The applications are endless! Pretty much all of grad level engineering classes, all of grad level Stats classes, all of grad level Econ classes include applications of LA. Heck quantitative research work in social sciences are still included via stats.
- selimthegrim 3y ago
  When I taught linear algebra I used Strang and a book from the UK by Morris that seems to be out of print these days.
tripdout 3y ago
Wow, we used this textbook (albeit alongside a more beginner focused one) in first year undergrad Computer Science.
anta40 3y ago
What's your recommendation material for first course? Strang, perhaps? I used Howard Anton's.
- mayd 3y ago
  I would definitely choose Strang to begin learning Linear Algebra. He develops the geometric intuition you need for applying it in other subjects such as Calculus or Statistics or Physics. Linear Algebra is not just Abstract Algebra, at least for most of us, including most mathematicians. After Strang, or even alongside him, if you want more rigour then you would benefit from a text like that of Anton, or Friedberg, or Curtis or, God forbid, Axler. FYI I bought Axler's book (3rd edition).

febra_ 3y ago

I have this book. It's been more than great. I fully recommend it to anyone looking for a quick and simple introduction to linear algebra

some_math_guy 3y ago

Like basically everybody else I teach out of this book, and I'm happy to see a new edition. I'm curious what's changed/added -- I already am unable to get through the whole thing in a semester. At our school students take a computational linear algebra course first (with a lot of row reduction). So I am slowed down a bit by constantly trying to help the students see that the material is really the same thing both times through. I do wish there were a little more of that in Axler.

blovescoffee 3y ago
I've studied from this book. Since you're teaching out of it, I'm curious if you've read/have an opinion on Strang's books. I love his lectures :)
- some_math_guy 3y ago
  Sure, I am very familiar with them -- I actually TAed 18.06 for Strang once upon a time. They're great books too. Which is better is mostly a question of what point of view you're after -- if you want to actually calculate anything, Axler's book is not going to help you, but if you want a more conceptual view of the subject it's best place. If you're really serious about learning linear algebra, you probably want to read both, first Strang, then Axler.
  - blovescoffee 3y ago
    That's awesome. I bet TAing for Strang was a great experience. Now I'm curious if you have any recommendations for "actually calculating something"

seo-speedwagon 3y ago

The 3rd edition of this book is what my undergraduate linear algebra course used. It was a fantastic book. I feel like more computation- and determinant-heavy approaches can make the subject feel like a slog, but this book made me really enjoy, appreciate, and get a gut-level understanding of the subject

gmiller123456 3y ago

It's not at all obvious from the headline, but the news is that the book is FREE, you can download the PDF from the first link.

cyberax 3y ago

I skimmed it, and it's a bit too succinct for my liking. I guess it's inevitable if you want to try to pack that much material into one book.

__rito__ 3y ago

This book is Open Access and you can download it from this link [0]. [0]: https://link.springer.com/content/pdf/10.1007/978-3-031-41026-0.pdf https://link.springer.com/content/pdf/10.1007/978-3-031-4102...

Alifatisk 3y ago
Amazing!

sieste 3y ago

The look on Christina of Sweden's face on page 1 made me laugh, she looks exactly how you would imagine a 17th century princess hearing about Linear Algebra.

nextos 3y ago

I dislike the typesetting changes that came after the second edition. It was a really elegant book, reminiscent of other classic Springer Undergraduate Mathematics Series tomes. Lots of distracting color, highlighting, and boxes were added, which IMHO make the book less clear. Obviously, the content is still great.

fiforpg 3y ago
Yep. Huge kudos to the author for making it available, but the PDF does feel sloppy with all the bright colors and images. In science textbooks, less is more. Compare this to the very latest edition of Stewart's calculus, which now uses even more pastel, subdued colors for diagrams.
- nextos 3y ago
  Exactly, a calm black and white design does not need to be unfriendly. Hubbard & Hubbard or MacKay are two examples of beginner-friendly books with great typesetting.
blt 3y ago
with the power of `tcolorbox` comes great responsibility.

readthenotes1 3y ago

Linear Algebra Done Right This Time We Promise (4th try) J/k ;)

selimthegrim 3y ago
4th sequence

navanchauhan 3y ago

For the course “Linear Algebra for Math Major”, we are using both LADR and Linear Algebra Done Wrong (Open Access as well)

mike986 3y ago

I have surveyed every LA books out there and a lot of amazons reviews claimed axler’s book is the best LA book. It might be for case for printed books for sale. But I stumbled upon Terrance Tao’s pdf LA lecture slides on his website and it is so much better than all the books I’ve surveyed. The writing is super clear and everything is built from the first principles. (BTW terry’s real analysis book did the same for me. Much more clear and easy to follow than the classics out there)

abdullahkhalids 3y ago
I believe these are notes that you are referring to https://terrytao.files.wordpress.com/2016/12/linear-algebra-notes.pdf https://terrytao.files.wordpress.com/2016/12/linear-algebra-...
- pstuart 3y ago
  Thanks!
- havercosine 3y ago
  These notes are excellent. One good thing is how often Terence Tao gives real life examples and analogies, contrary to what one may expect from a fields medal winner. From utilitarian perspective, reading Axler's book looks like comically bad use of one's time.
zeroonetwothree 3y ago
I'm not sure that Axler's book is great as a first LA book. I would go with something more traditional like Strang. Although I really didn't feel like I "got" LA until I learned algebra (via Artin). By itself LA feels very "cookbook-y", like just a random set of unrelated things. Whereas in the context of algebra it really makes a lot more sense.
- mohamez 3y ago
  >I'm not sure that Axler's book is great as a first LA book. Linear Algebra Done Right is a text for beginners who want to study linear algebra in a proof based, mathematically rigorous way. So, if you want that I think it's a good fit as a first linear algebra book.

fiforpg 3y ago

About avoiding determinants to the degree that this book does: while I agree it makes sense to delay introducing them, the goal should not be avoidance but clarity. The way author has to bend himself backwards here when dealing with eigenvalues isn't great either. I would recommend Strang for a healthy balance in handling determinants.

jjoonathan 3y ago
I read Strang and then Axler. Strang is great at numerics but weak at presenting the abstract picture. I feel like if I had taken, say, finite elements (or any other subject where it's important to take the abstract / infinite dimensional picture seriously before reducing to finite dimensions) right after Strang without reading LADR then I'd have been seriously underprepared.
- fiforpg 3y ago
  You have a point in that to understand any particular subject well, it makes sense to read more than one book on it, at least to compare the different perspectives. Also worth noting that Strang has a couple of similar linear algebra books, so we might not even be discussing the same text.
  - jjoonathan 3y ago
    That's entirely possible, but in the context of introductory books I think it's fair to assume & limit scope to Strang's "Introduction to Linear Algebra" and Axler's "Linear Algebra Done Right." I am an applications-oriented person and my inclination was to go directly from a matrix/determinant heavy picture into applications. Strang['s intro text] only. I am extremely glad that someone intercepted me and made me get some practice with abstract vector spaces, operators, and inner product spaces first, using Axler. This practice bailed me out and differentiated me from peers on a number of occasions, so I want to pass down the recommendation.
- nerdponx 3y ago
  FWIW I think this is the benefit of Strang. If you're in science or engineering or statistics, often you don't need the general picture, and IMO too much generality gets in the way of understanding. Start with a good understanding of the most important cases that appear in applied work, and drill them until you're fluent with them. Then generalizing will be easier.

imjonse 3y ago

From the preface. "You cannot read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast." Sadly, he's probably right.

zeroonetwothree 3y ago
Meh, there's different goals you could have. I actually find it enjoyable to read math more quickly (almost like a novel) which gives you a good sense of a lot of the higher-level themes and ideas. Then if it's interesting I might spend more time on it.
- mohamez 3y ago
  He is talking about reading it after you decided that it is interesting.
mohamez 3y ago
Tips on Reading Mathematics[1]: - Be an active reader. Open to the page you need to read, get out some paper and a pencil. - If notation is defined, make sure you know what it means. Your pencil and paper should come in handy here. - Look up the definitions of all words that you do not understand. - Read the statement of the theorem, corollary, lemma, or example. Can you work through the details of the proof by yourself? Try. Even if it feels like you are making no progress, you are gaining a better understanding of what you need to do. - Once you truly understand the statement of what is to be proven, you may still have trouble reading the proof—even someone’s well-written, clear, concise proof. Try to get the overall idea of what the author is doing, and then try (again) to prove it yourself. - If a theorem is quoted in a proof and you don’t know what it is, look it up. Check that the hypotheses apply, and that the conclusion is what the author claims it is. - Don’t expect to go quickly. You need to get the overall idea as well as the details. This takes time. - If you are reading a fairly long proof, try doing it in bits. - If you can’t figure out what the author is doing, try to (if appropriate) choose a more specific case and work through the argument for that specific case. - Draw a picture, if appropriate. - If you really can’t get it, do what comes naturally—put the book down and come back to it later. - You might want to take this time to read similar proofs or some examples. - After reading a theorem, see if you can restate it. Make sure you know what the theorem says, what it applies to, and what it does not apply to. - After you read the proof, try to outline the technique and main idea the author used. Try to explain it to a willing listener. If you can’t do this without looking back at the proof, you probably didn’t fully understand the proof. Read it again. - Can you prove anything else using a similar proof? Does the proof remind you of something else? - - What are the limits of this proof? This theorem? - If your teacher is following a book, read over the proofs before you go to class. You’ll be glad you did. [1] Reading, Writing, and Proving: A Closer Look at Mathematics By Ulrich Daepp and Pamela Gorkin.

photochemsyn 3y ago

This is linear algebra for undergraduate math majors, but if you just want an basic understanding of the topic with a focus on computational applications, Poole's "Linear Algebra: A Modern Introduction" is probably more suitable as it's heavy on applications, such as Markov chains, error-correcting codes, spatiel orientation in robotics, GPS calculations, etc. https://www.physicsforums.com/threads/linear-algebra-a-modern-introduction-by-david-poole.665360/ https://www.physicsforums.com/threads/linear-algebra-a-moder...

HybridCurve 3y ago

If you think a textbook is good, the digital edition should available for review as they will often be invaluable in hardcopy for continued reference. I am pleased the author adheres in some part to this mode of thinking.

Buttons840 3y ago

I've been waiting for this for 6th months. Thanks to Sheldon Axler for making it available for free. This is intended to be a second book on Linear Algebra. For a first book I suggest "Linear Algebra: Theory, Intuition, Code" by Mike X Cohen. It's a bit different than a typical math textbook, it has more focus on conversational explanations using words, although the book does have plenty of proofs as well. The book also has a lot of code examples, which I didn't do, but I did appreciate the discussions related to computing; for example, the book explains that several calculations that can be done by hand are numerically unstable when done on computers (those darn floats are tricky). For the HN crowd, this is the right focus, math for the sake of computing, rather than math for the sake of math. One insight I gained from the book was the 4 different perspectives of matrix multiplication. I had never encountered this, not even in the oft-suggested "Essence of Linear Algebra" YouTube series. Everything I had seen explained only one of the 4 views, and then I'd encounter a calculation that was better understood by another view and would be confused. It still bends my mind to think all these different perspectives describe the same calculation, they're just different ways of interpreting it. At the risk of spamming a bit, I'll put my notes here, because this is something I've never seen written down elsewhere. The book has more explanation, these are just my condensed notes. 4 perspectives on matrix multiplication ======================================= 1 Element perspective (all possible dot / inner products) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (row count × row length) × (column length × column count) In AB, every element is the dot product of the corresponding row of A and column of B. The rows in A are the same length as the columns in B and thus have dot products. 2 Layer perspective (sum of outer product layers) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (column length × column count) × (row count × row length) AB is the sum of every outer product between the corresponding columns in A and rows in B. The column count in A is the same as the row count in B, thus the columns and rows pair up exactly for the outer product operation. The outer product does not require vectors to be the same length. 3 Column perspective (weighted sums / linear combinations) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (column length × column count) × (column length × column count) In AB, every column is a weighted sum of the columns in A; the weights come from the columns in B. The weight count in the columns of B must match the column count in A. 4 Row perspective (weighted sums / linear combinations) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (row count × row length) × (row count × row length) In AB, every row is a weighted sum of the rows in B; the weights come from the rows in A. The weight count in the rows of A must match the row count in B.

mohamez 3y ago

Linear Algebra Done Right is a good book for people who want to study the subject of linear algebra in a proof based, mathematically rigorous way. Here [1] you can find Sheldon Axler himself explaining the topics of the book in his YouTube channel! How wonderful is that! Here [2] you can find the solutions to the exercises in the book. [1] https://www.youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmSinFV0wEir2Vw https://www.youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmS... [2] http://linearalgebras.com/ http://linearalgebras.com/

harry8 3y ago
> Here [2] you can find the solutions to the exercises in the book. > >[2] http://linearalgebras.com/ http://linearalgebras.com/ I have trouble with believing an author has the students' interests at heart when the answers to solutions are not /in/ the book. Why on earth not? It makes no sense from any student perspective. I know it's common in textbooks to not even have them available at all. Any ideas why these are the norm not putting them where they clearly belong? It just seems so hostile.
- layer8 3y ago
  It’s too easy to just look up the solution instead of being forced to think hard about the problem. This is training for when you’ll later encounter problems outside of textbooks where you’ll have no choice but to solve them on your own. And you’re supposed to have teaching assistants or similar available when you really remain stuck.
  - harry8 3y ago
    Yeah sorry that's just total and utter BS. No sale. I am an adult. >99% of students of university subjects are adults. Even assuming your premise totally, anyone so incredibly stupid they can't have it explained in the text when is the right time to use the solutions is too stupid to learn from the text. Axler puts them on a website that is less than 30 seconds away. "No choice..." Describes what proportion of texbook sales?

ayakang31415 3y ago

This is a great book, but you must first be familiar with proof-focused mathematics (logic and set theory). If you are not, I first suggest you study a book called "How to Prove It: A Structured Approach" by Daniel J. Velleman before studying LA Done Right.

noqc 3y ago
You need a setting in which to learn proof based mathematics, and linear algebra really is the first place where students are ready for that journey. Not everyone is going to be able to do it, but it's very incorrect to say that you must be familiar with proof. One must start somewhere, and ZF ain't it.
- ayakang31415 3y ago
  Sorry, what is ZF?
  - noqc 3y ago
    Zermelo-Fraenkel, the axioms of set theory (minus choice). This would be what you were teaching if you were to start teaching mathematics by teaching its (classical) foundations.
    - ayakang31415 3y ago
      I briefly checked the axioms, and if I recall correctly, Velleman's book does not delve into ZF in detail, but it is a beginner friendly book to give enough exposure to quantitative logic and set theory to feel comfortable working on proof-based mathematics such as LA (not survey version of LA, which should be called matrix algebra instead).
      - noqc 3y ago
        Sorry, I wasn't talking about Velleman's book on whatever, I was just challenging the sentiment that you should learn the "basics of proof based math" before linear algebra. Linear algebra is the basics of proof based math. Edit: I should clarify that I believe that there are many approaches to learning mathematics, I'm sure that Velleman's book is fine, just that it isn't true that you should not try to learn from Axler, although perhaps someone who has a bit less of an axe to grind.

ekm2 3y ago

This is good as a second course on Linear Algebra.For a first course,use (I am not kidding) Linear Algebra Done Wrong by Sergei Treil https://www.math.brown.edu/streil/papers/LADW/LADW.html https://www.math.brown.edu/streil/papers/LADW/LADW.html

endymi0n 3y ago
omg just had a look and this one is just everything I hate about mathematics and academia. Starts with lots of random definitions, remarks, axioms and introducing new sign language while completely disregarding introducing what it‘s supposed to do, explain or help with. All self-aggrandization by creating complexity, zero intuition and simplification. Isn‘t there anybody close to the Feynman of Linear Algebra?
- ekm2 3y ago
  There is no royal road bro
  - 77pt77 3y ago
    In his mind his above royals, so a royal road would be demeaning to him.
- PartiallyTyped 3y ago
  What about Axler’s then?
- gadrev 3y ago
  Gilbert Strang's course on Linear Algebra. Playlist: https://www.youtube.com/playlist?list=PL49CF3715CB9EF31D https://www.youtube.com/playlist?list=PL49CF3715CB9EF31D Not as big in scope, though, but great introduction.
- bscphil 3y ago
  Yeah, a good example is on the second page of the first chapter: > Remark. It is easy to prove that zero vector 0 is unique, and that given v ∈ V its additive inverse −v is also unique. The is the first time the word "unique" is used in the text. Students are going to have no idea whether this is meant in some technical sense or just conventional English. One can imagine various meanings, but that doesn't substitute for real understanding. This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. On the surface the opposite is true - you complain, for instance, that the text jumps immediately into using technical language without any prior introduction or intuition building. My take is that intuition building doesn't need to replace or preface the use of formal precision, but that what is needed is to bridge concepts the student already understands and has intuition for to the new concept that the student is to learn. In terms of intuition building, I think it's probably best to introduce vectors via talking about Euclidean space - which gives the student the possibility of using their physical intuitions. The student should build intuition for how and why vector space "axioms" hold by learning that fundamental operations like addition (which they already grasp) are being extended to vectors in Euclidean space. They already instinctively understand the axiomatic properties being introduced, it's just that the raw technical language being thrown at them fails to connect to any concept they already possess.

tenderfault 3y ago

Right. “Complex numbers were invented so that we can take square roots of negative numbers” Why? I wish a textbook tell me why, right there.

jjoonathan 3y ago
Because complex numbers make the fundamental theorem of Algebra nice and simple rather than complicated and ugly. In turn, this makes the spectral theorem of linear algebra nice and simple rather than complicated and ugly. In turn, this makes a bunch of downstream applications nice and simple rather than complicated and ugly. You will get a feel for this if you work Axler's problems. More importantly, you will gain an intuition for the fact that if you turn up your nose at complex numbers while going into these application spaces, you are likely to painstakingly reinvent them except harder, more ugly, and worse. Example: in physics, oscillation and waves A. underpin everything and B. involve energy sloshing between two buckets. Kinetic and potential. Electric and magnetic. Pressure and velocity. These become real and imaginary (or imaginary and real, it's arbitrary). This is where complex numbers -- where you have two choices of units -- absolutely shine. Where you would have needed two coupled equations with lots of sin(), cos(), trig identities, and perhaps even bifurcated domains you now have one simple equation with exponentials and lots of mathematical power tools immediately available. Complex numbers are a huge upgrade, and that's why anything to do with waves will have them absolutely everywhere.
- wwarner 3y ago
  You might think that in every real world application, complex numbers are introduced as a convenience, and that every calculation that takes advantage of them ends with taking the real part of the result, but that's not the case. In QM, the final answer contains an imaginary part that cannot be removed.
  - BeetleB 3y ago
    Having studied QM, I disagree. You can always introduce trigonometry and remove it. Keep in mind that everything observable is a real number even if the intermediate calculations involve imaginary numbers.

MrBlueIncognito 3y ago

I don’t know if it’s just me, but I’m terribly lost in the search for the right texts. Every time I come across a new book/resource, it compounds the confusion. I find myself incapable of sitting down with a book and working through it without switching to another book in-between. I’d be really happy to hear if anyone has found a solution to this unproductive but sticky habit.

generationP 3y ago

4th edition and still hasn't bothered to disambiguate between polynomial and polynomial function. Still a good example of mathematical writing. But, as Einstein supposedly said, "as simple as possible but not simpler". Why is it always American authors that forget that last part?

beyonddream 3y ago
Do you suggest any book that does disambiguate between polynomial and polynomial function (i thought they were equivalent!) ?
- adrian_b 3y ago
  I do not know what the previous poster has meant to say, but polynomials can be defined in a more abstract way as just elements of certain vector spaces on which an additional operation of multiplication between polynomials is defined, which obeys special rules (the "carry-less multiplication", which is implemented in hardware by most modern CPUs, is an example of polynomial multiplication). In computers there are many applications of such abstract polynomials, e.g. for error detecting or correcting codes, pseudo-random number generation, authenticated encryption and others, which depend only on the rules for addition, internal multiplication and multiplication with scalars, and which have nothing to do with the polynomial functions associated with the polynomials.

slowhadoken 3y ago

the good old indispensable and underappreciated linear algebra

NoNameHaveI 3y ago

To the author: I commend you on your work! As a former full-time educator, I believe that one of the best thing you can do to promote usage is to create supplementary materials (lecture slides, quiz deck in Blackboard format, etc. I do realize there are differences between everyone, but when I was an educator, there were many times when I was expected to teach multiple subjects that I had little to no experience with, with as little as one days notice in many cases (the adjunct had a baby. Can you teach this?). TLDR; Having a good text is a good start. To increase adoption and usage, create teaching materials, which saves valuable time for instructors. Peace out

nixer6872 3y ago

Just read Dummit Foote. L.A. Done Right is not that good as its fancy title.

arnonejoe 3y ago

The linear algebra refresher course on Udacity is excellent (and free). https://www.udacity.com/course/linear-algebra-refresher-course--ud953 https://www.udacity.com/course/linear-algebra-refresher-cour...

deleted 3y ago

[deleted]

magnio 3y ago

A lot of the discordance in this thread can be boil down to "different people having different goals when reading". As for Linear Algebra, due to its wide relevance, it is hard for one book to accommodate to all needs of people from different fields and levels of mathematical maturity. I would say there are 2 main reasons for one to study LA: - To understand and use its toolbox in practice: in fields like computer graphics, engineering, statistics, data analysis, modeling, etc. - To understand and use its toolbox in other fields of (pure) math: single, multivariable, and functional analysis, abstract algebra, etc. (Sometimes the latter goal is coupled with a secondary goal of introducing the student to mathematical proof. However, for the self-learners, I don't think using a LA book is good for learning how to write proof for the first time.) Unlike in calculus where Spivak serves as the one-size-fits-all, readers should identify their reasons before picking an LA book. I have not much experience with the first goal. Strang's book seems quite popular, and given the title, I would say it fits this goal. Axler's book falls squarely into the second goal, and I wager it's a good book for this purpose if not for two things: his avoidance of determinants and the dearth of any computational aspects or applications (yes, even theoretical LA books ought to be grounded in reality). There are three books that, in my opinion, do better than Axler here: - "Linear Algebra" by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence: dry, does not instill in me the sense of excitement that Axler does. However, it is comprehensive and rigorous, with a decent amount of explanation and a good selection of exercises. Application aspects are present but quite disparate and incoherent. This is the old and long-standing recommendation. - "Linear Algebra" by Elizabeth S. Meckes and Mark W. Meckes: a new book that somehow manages to combine the material of Friedberg and the eloquence of Axler. The rigor and exercises are comparable to Friedberg, but the applications are presented much better here. - "Linear Algebra" by Sterling K. Berberian: a little bit more abstract than the other two, but the exposition is still very clear. It connects LA to other fields of math and gives plenty of examples and motivations (shocking I know). The exercises department is good. Honorable mentions to the classics of Serge Lang and Paul Halmos. They are concise and, as with the rest of their works, wonderfully written. For "Linear Algebra Done Wrong", well, I feel like it's a stripped down Friedberg: shorter but not better. (The proof of the multiplicativity of determinant is just some hand-waving at "a lucky coincidence".)

bigbacaloa 3y ago

[dead]

akasakahakada 3y ago

Seriously this is the worst book ever in the textbook history. People always laugh for stuff like "tensor is stuff that works like a tensor", but this book is literally doing that in every page.

bsaul 3y ago

Any advice for calculus ? something that would take you from derivatives up to the kind of advanced analysis required in quantum field theory ?

lysecret 3y ago

In my opinion the best course on LA by far: https://youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS14F4NxAZudSmv&si=qDiayasMQw7P-sJ_ https://youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS14F4N...

Ste_Tokyo 3y ago

The book starts out writing vectors as row vectors (1 x n matrices), then later introduces the matrix of a vector as a column vector. Just represent vectors as column vectors from the start, define R^n as R^n,1 (in the notation of the book) and then there's no confusion and you get matrix multiplication of column vectors as students learned in high school: M.x where M is an m x n matrix and x is an n x 1 column vector, with product an m x 1 column vector. Stick to this all the way through a first linear algebra textbook and undergrad tears can be avoided. e.g. example 3.32 in the text: T(1,0) = (1,2,7). Just write these as column vectors and the matrix multiplication is easier to visualize. (Also, T acting on a real number was written T(x) with brackets so T acting on a vector should really be written T((1,0)). Inconsistencies in notation like this lead to pain.) Then, for example, when you want to show that the rows of a matrix are independent, you can use a row vector to write succinctly x.M = 0 => x = 0 where x is now a row vector. And it's clear we're doing something other than transforming space vectors here as we've used column vectors for vectors representing points in space. For undergrad textbooks you want to decide on a convention, one that matches what was done in high school preferably, and stick to it.

uxp8u61q 3y ago
You do realize it makes no difference whether the vectors are written horizontally or vertically? Part of mathematical maturity is not getting hung up on pointless notation like this.
- Ste_Tokyo 3y ago
  Clearly the goal of a first course in linear algebra is not mathematical maturity. If you don't set up the notation consistently from the start, once you start introducing change of bases formulae or proving row rank = column rank, etc., some students will get confused. See all the questions from confused students on Maths Stack Exchange about this point. The way this particular author has handled the issue is perverse: first he writes vectors as row vectors, then he introduces a mapping M(v) which maps the vector v to its representation as a column vector. Also, as I tangentially noted, he misses out the brackets for a function writing T(0,1) instead of T((0,1)), which is just incompetent. The book should be titled "Algebra Done Shite" not "Algebra Done Right" ;)

nimish 3y ago

The elusive linear algebra book that isn't simply matrix algebra. A gem.

ivzhh 3y ago

This book and mathematical analysis from Zorich are my favorite two math books.

RedCinnabar 3y ago

I know this book is not intended to be used as a first reading on Linear Algebra but, to me, this text isn’t good even for a second(or a third) read. The author hurries up too much on certain parts. I think that Serge Lang’s Linear Algebra does a better job in explaining pretty much every topic of the subject.

allangrant 3y ago

I. X

anjanb 3y ago

There's so much discussion here but I couldn't decide what is a good first course/book on linear algebra. Can someone please summarize and share a couple of books/courses ? Thanks

fattah25 3y ago

Everything good if it's free.