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I’ve tried to make my own very general roadmap to category theory based on the questions asked on MSE and MO regarding self-studying this area. It’s something like this:

  • Abstract Algebra
  • Point Set Topology
  • Algebraic Topology
  • Category Theory

Now, I read somewhere recently that abstract algebra should be done after linear algebra. But I’ve started on Dummit and Foote and found it quite easy and readable, despite the extent of my linear algebra knowledge being the set-theoretic definitions of vector spaces and modules, what linear independence is, and how to multiply matrices.

So my question: Given that I’m not currently interested in studying linear algebra for the sake of linear algebra, should I study it or just keep going with abstract algebra? If I should study it, what books should I look at?

Also: Applied math is not my goal. That being said, examples help when introducing an abstract concept. It’s like how knowing what a group is might help when defining $\mathcal{L}$-structures, but while learning logic it need not be applied to group theory.

P-addict
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    Well, when you get to modules over rings you will have to do some linear algebra anyway. When you get to fields extensions, you will also have to do some linear algebra. Who knows, it is possible to catch up whatever you missed along the way. – MoonLightSyzygy Jan 20 '20 at 21:57
  • You wish to to see: Why Study Linear Algebra. As for book recommendations, please see the dozens of LA book questions already asked. – Brian61354270 Jan 20 '20 at 21:58
  • Why do you want tout do learn Category Theory? How did you even know that it exists but you are not familiar with linear algebra. (No judgement, I am just curious) – klirk Jan 20 '20 at 21:58
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    You listed Algebraic Topology. I'd be amazed if you can do much algebraic topology without knowing linear algebra. You might squeeze by up through the theory of fundamental groups. But beyond that, the theory and computation of homology groups requires solid knowledge of linear algebra. I've seen too many students who can't compute a basis for the image or kernel of a linear transformation and who therefore fall by the wayside when studying homology groups. – Lee Mosher Jan 20 '20 at 22:04
  • @Brian Thank you, I will look more at those. I think I have motivation to study linear algebra, though. I guess the reason I chose to ask this question is because I don't want to read a 500-page textbook on linear algebra if I'm only going to use three of those chapters. – P-addict Jan 20 '20 at 22:06
  • @klirk I want to learn CT because it looks interesting, I am a big fan of pure math and "abstract, general stuff" (I like set theory, for example), and intuitionist logic fascinates me. But apparently things like topoi of sheaves of sets on a topological space are quite important in studying this area. And I know it exists because MSE. – P-addict Jan 20 '20 at 22:07
  • @LeeMosher Ok, thanks for letting me know. I will look more into linear algebra, then. – P-addict Jan 20 '20 at 22:10
  • Maybe get a copy of Theory of Modules (see here also) by Alexandru Solian. See my comments about Solian's book here. Incidentally, I believe I have a .pdf file of this book (on my other computer, now off for the day, so I don't want to check right now), but I can't find one on the internet that appears to be freely available (but I also haven't looked very hard for such a copy). I also have an essentially new hardback copy of the book . . . – Dave L. Renfro Jul 20 '20 at 19:02
  • I agree with the suggestion of Artin for someone starting out. Abstract algebra and linear algebra enrich each other and should ideally be studied together by those who are primarily interested in mathematics as opposed to its applications (where one or the other may be more urgent, depending on the field). However, the typical arrangement in U.S. universities in which linear algebra comes first does have the effect that linear algebra is neglected in many algebra books. Now, you may already be well into D&F, and in this case it might not make sense to start over again with a new... – Anonymous Jul 20 '20 at 23:08
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    undergraduate algebra book. In that case an alternative would be to read the linear algebra chapters (mainly Chapters 13-15) of Lang's Algebra. That will fill in any gaps, and it's written at an appropriate level of sophistication for someone who's read much of D&F. You may also want to look at a book with problems in linear algebra such as the one by Proskuryakov. Alternatively, you might want to read a book like Advanced Linear Algebra by Roman, or else an introductory book at a relatively high level of difficulty, such as Linear Algebra and Geometry by Manin and Kostrikin. – Anonymous Jul 20 '20 at 23:21
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    A good book intended for a quick second pass through linear algebra is Algèbre linéaire et groupes classiques by Bertin and Bertin. There is an accompanying problem book by Malliavin and Warusfel. – Anonymous Jul 20 '20 at 23:47

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Linear algebra is very important. I disagree that it is a prerequisite for abstract algebra, but you definitely need to know it pretty well because it shows up everywhere. You can keep doing what you're doing, but perhaps also look at Artin which includes linear algebra in the study of abstract algebra.

Glorfindel
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Matt Samuel
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  • I was just about to post an answer saying the exact same thing. – user729424 Jan 20 '20 at 21:59
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    Also, just to be clear "Artin" is the book Algebra written by Michael Artin. It's excellent, and it will go over all the linear algebra that you need to know. Also, it will cover these topics in a way that will be maximally useful for you, given what your goals are. – user729424 Jan 20 '20 at 22:02