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As the title implies, I'm in need of some guidance from people with experience in linear algebra on how I should approach the subject to learn more effectively.

My background: I'm a masters student in computer science working with theoretical stuff that involves lots of linear algebra. Before the masters, my education in mathematics was, to be honest, terrible. Fortunately, I have being able to keep up and have taught myself a fair amount of discrete mathematics. As a result, today I feel much more comfortable with proof writing (or the so called mathematical maturity). Now, the quest is to learn linear algebra.

My goal: Learn (or master, if possible) linear algebra until the end of this year. I need to learn well at least the basics (up to orthogonality, projection matrices, etc) until September. Although I'm not a beginner in the subject, I'm very far from proficient. I'm interested in learning linear algebra for further studies in theoretical computer science, e.g. algorithms, graph theory, combinatorial optimization, etc. Let me make some points clear:

  • I don't work with numerical linear algebra, and don't intend to.
  • I'm NOT interested in learning linear algebra for machine learning and/or computer graphics (although I find ML interesting..). Just saying because people in CS usually associate linear algebra with these subjects.
  • I'm looking to improve my mathematical maturity, so a proof-based approach to learn linear algebra is very welcome.

My resources: I have at my disposal three books: Axler's Linear Algebra Done Right, Strang's Introduction to Linear Algebra, and Schaum's Outline of Linear Algebra. A while ago, I started reading Axler's and found it very interesting, but I didn't even finish chapter 1 because I have some concerns lurking in my head. I feel I'm skipping steps and jumping right into an advanced book. Here are my concerns about each book:

  • Strang: as far as I could see, the book deals almost exclusively with matrices and applications. Vector spaces are treated with less rigor, and linear transformations appear in a brief chapter towards the end of the book. Although I think the appeal for intuition seems nice, the lack of rigor in many parts bothers me a lot. I feel I'd be missing important details. In addition, the book is filled with applications that don't interest me at all (e.g. differential equations, circuits, networks, etc). A huge plus is the online lectures though. What worries me the most is the lack of rigor (no proofs) and emphasis in applications that don't interest me.
  • Axler: beautifully written and very readable, with extensive treatment of vector spaces, linear transformations, etc. However, I saw almost no matrix algebra and some topics seem very out of my context (e.g. vector space of polynomials). Nevertheless, I genuinely find the book very interesting, but at this moment I have to make pragmatic decisions and leave less useful stuff to be seen later. A huge plus of this book is that I'd have the opportunity to improve my proof-writing skills and lay a solid foundation. What worries me the most is the absence of matrix algebra.
  • Schaum: just a reference with a bunch of exercises. Don't know if it's sufficient by itself. I thought in coupling this one with Axler's.

It's like Axler and Strang are two extremes, one treating something the other does not. Strang starts with systems of linear equation with those endless and boring mechanical calculations, but seems to have an extensive treatment of matrix algebra. On the other hand, Axler jumps right into the interesting stuff (vector spaces), but has almost nothing about matrices.

The ideal approach, of course, would be to read both in sequence. But I'm afraid time will not be my friend here and (forgive me for the drama) I'm desperate to learn this subject. Nevertheless, if it's really worth the time and there's no better way, I think I'd be able to manage it. Do you think it'd be worth it? Should I look for another text that has the best of both worlds? Or maybe you have alternative suggestions that would be more effective?

Lucas Peres
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  • Read Strang and then Axler, my man. Don't let your concerns hold you back. – littleO Jun 08 '21 at 06:34
  • Hoffman and Kunze's Linear Algebra is well-written and geared more towards theory instead of applications, so it seems aligned with your interests, although, I second @littleO suggestion as well. – C Squared Jun 08 '21 at 06:44
  • Axler's book is nice, but it's intended as a second course and it is certainly not the most conventional treatment of some topics. As a possible alternative, check out Linear Algebra Done Wrong if you haven't already. It's free, so no harm done if you don't like it. Hoffman and Kunze is good but also probably better as a second course. I also find it very dry. –  Jun 08 '21 at 06:49
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    a word of caution -- to my knowledge all 3 of those books only treat $\mathbb R$ and $\mathbb C$ as fields (with at most a couple sentences indicating LA works over other fields though the reader won't know what this means). But in CS, e.g. information and coding theory, algebraic combinatorics as well as elsewhere, other fields are sometimes used, esp $\mathbb F_2$. A lot of people are not able to make the conceptual jump. One approach would be to ignore all those books and instead do e.g. Strang's MIT OCW course, and then go through the first 4 chapters of Artin's Algebra. – user8675309 Jun 08 '21 at 18:16
  • @user8675309 After taking a look in Artin's Algebra, I understood your concern. In the long-run, I think you are right, I do want to study Algebra eventually. But in fact my current work involves only the $\mathbb{R}$ field, as treated in any linear algebra book. For now, after reading the comments, I'm tempted to read Axler's coupled with something else (e.g. Linear Algebra Done Wrong maybe?) or choose an entirely different book. I'm considering dropping Strang altogether. – Lucas Peres Jun 08 '21 at 18:48
  • Some people recommended me Friendberg's Linear Algebra, which seems very promising, but unfortunately I don't have access to a physical copy. :( – Lucas Peres Jun 08 '21 at 18:51
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    I'm also a fan of Linear Algebra by Friedberg, Insel, and Spence. It gives a solid, clear, standard presentation of linear algebra. If you don't read Strang, be sure to learn the various ways of thinking about matrix multiplication that Strang emphasizes. For example, $Ax$ is a linear combination of the columns of $A$. Multiplying by $A^{-1}$ changes basis to the basis consisting of the columns of $A$. And be sure to learn the "four subspaces" picture that Strang also emphasizes, and the related material about least squares problems. – littleO Jun 08 '21 at 19:41
  • @littleO Nice! Thanks for pointing out this aspect of Strang. :) – Lucas Peres Jun 08 '21 at 21:11

2 Answers2

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Hoffman Kunze's is a good book.I highly recommend this book.Axler's book doesn't cover the all syllabus of linear Algebra. Axler skipped many important topics of linear Algebra.It's easy to waste time with a poorly written textbook.

"Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty"- Archimedes

Mathematics is not a competitive sport -Grothendieck

It’s also good to remember that professional mathematics is not a sport (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest “score”, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications. For these tasks, mathematics needs all the good people it can get.- Terence Tao

jasmine
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Based on your remarks, I'd say make your focus Axler, or some other similarly theoretical book, and spend $\textit{some}$ time studying the matrix stuff if you feel you need more (I don't know to what extent matrix computations are important for your work). Alternatively, you can first skim through Strang to remind yourself of the subject or acquaint yourself with the basics, and then begin the more serious treatment of Axler. But the bulk should undoubtedly (in my humble opinion) be from a book similar in character to Axler. Linear algebra isn't about matrices, it's about (finite-dimensional) vector spaces and linear transformations. The matrix computations you can, after a certain foundational understanding, just pick up real quick when you need them. You can't just pick up a solid understanding of vector spaces real quick.

Stuck
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  • Thank you for the advices. Speaking of theoretical books, do you have any thoughts on Axler's approach? I've heard many times that the teaches the subject in a peculiar way (e.g. no determinants). Does it harm the learning experience or further studies in mathematics? – Lucas Peres Jun 08 '21 at 16:26
  • @LucasPeres Axler's book doesn't cover the all syllabus of linear Algebra. Axler skipped many important topics of linear Algebra.It's easy to waste time with a poorly written textbook. – jasmine Jun 08 '21 at 16:34
  • @jasmine That worries me. But do you think what is left out would be essential for a first course? – Lucas Peres Jun 08 '21 at 16:55
  • @LucasPeres There is no first course in linear Algebra .One good book is sufficeint .You have to love mathematics without any expectation otherwise you will not understand deeply .Try to read all topics of linear Algebra (Hoffman kunz) book ."Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty"- Archimedes...Mathematics is not a competitive sport -Grothendieck – jasmine Jun 08 '21 at 17:06