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Note : I ask this question from the point of view of a Pure Mathematics (no applications to Physics, engineering etc. here)

I currently have copies of Linear Algebra Done Right by Axler, and Abstract Algebra by Dummit & Foote. I am really eager to get started with Abstract Algebra, but I haven't gone through Linear Algebra fully rigorously with proofs yet (I've read through the whole of Strang's Introduction to Linear Algebra, which is mostly just an applications based book)

In an ideal world I would like to read through both of time thoroughly, but I currently do not have the time to (and Dummit and Foote is itself a monstrous book). So currently I am thinking that I would read Dummit and Foote rigorously and skim through Axler and fill in any possible gaps as needed.

Am I missing out much by taking this path? I know that the links between Linear Algebra and Abstract Algebra are mainly in the generalization of vector spaces to modules, and that Linear Maps and Linear Operators are present nearly everywhere throughout Pure Math, but apart from that I don't see much of Linear Algebra in higher mathematics, and I figured that those important parts (like Linear Maps/Operators) can be filled in as I go along through Abstract Algebra.

My intended direction is to get myself to a level where I can study Algebraic Topology, Algebraic Geometry, Differential Topology/Geometry, and I'm currently reading through Principles of Mathematical Analysis by Rudin and Topology : A First Course by Munkres. Obviously once I complete these books I will take a look at Graduate level books.

If you feel my approach is something that should be avoided, please let me know why, apart from the fact that in most universities a rigorous (if there even exists one) course in Linear Algebra comes before Abstract Algebra.

J. W. Tanner
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Perturbative
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    Having a time constraint is not a good thing. I believe having a reasonable good foundation of linear algebra is important. Now you can skip the time needed to go through Axler and spend all the time on Abstract Algebra except...you may need more time to comprehend certain linear algebra related concepts. That time could have been saved had you spent time on linear algebra first. So my take on it: Study vector spaces and eigenvalues/eigenvectors before moving ahead. It is NOT a waste of time. Good luck – imranfat Nov 07 '16 at 16:16
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    I took courses both on Linear Algebra and Abstract Algebra; Linear was first. I did find it useful to have had Linear Algebra before, because it meant I could give practical examples for plenty of things already, even though if I had had Abstract Algebra first, some of the proofs of theorems of linear algebra would've been trivial. That being said, I found it more important to have had Linear Algebra first, because it helped the Abstract Algebra sink in. – RGS Nov 07 '16 at 16:17
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    Linear Algebra gives you often a nice bunch of examples for abstract algebra. If you tend to Geometry, you should at least take a look at dual spaces. – ctst Nov 07 '16 at 16:17
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    Linear algebra is absolutely indispensable in the study of mathematics. However, if you have entirely read and understood Strang's Introduction to Linear Algebra, you are presumably decently well-prepared to move on to a first course in abstract algebra. – Mees de Vries Nov 07 '16 at 16:19
  • It is certainly possible to do. There will be a few times where an example or an exercise will use a vector space. However, if you have the basic understanding, that should be sufficient.
    I don't suggest you dismiss Linear Algebra so quickly, though. It seems to come up again and again in every high-level course.     – Doug M Nov 07 '16 at 16:24
  • Short answer: no, it is not needed. In fact I prefer a lot more abstract algebra, more funny, than the linear one. –  Nov 07 '16 at 16:24
  • @Doug M, could you elaborate specifically which aspects seem to pop up again and again? – Perturbative Nov 07 '16 at 16:28
  • @Perturbative. This might help you: http://math.stackexchange.com/questions/256682/why-study-linear-algebra And ignore Masacroso's answer... – imranfat Nov 07 '16 at 16:35
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    Well I think linear algebra is necessary for studying abstract algebra. Not only because it gives many examples in group or ring theory, also because it serves to study module and field theory. Also, if you want to study representation theory or differential geometry you definitely need to understand well linear algebra. – Xam Nov 07 '16 at 16:35
  • Pretty much the same question: https://math.stackexchange.com/q/2470576/29335 – rschwieb Apr 09 '18 at 15:22
  • @Perturbative Just curious, what did you end up doing and how did it turn out? I am also an undergraduate and just learned some Abstract Algebra from Dummit & Foote and some Linear Algebra from another book. – Ovi Nov 23 '18 at 21:09
  • @Ovi I ended up learning bits and pieces of Abstract Algebra from Dummit and Foote (and other abstract algebra books), and I'd say that the knowledge I gained from learning abstract algebra is actually what made some (maybe even most) aspects of linear algebra make sense to me. I'd say that these days if I need to look up a linear algebraic theorem or proof it's far easier for me to follow thanks to studying abstract algebra. To this day I haven't fully gone through a linear algebra textbook since I haven't really had the need too, whatever I need I can look up in a sort of piecemeal way. – Perturbative Nov 23 '18 at 21:39
  • I will say that learning abstract algebra (e.g group theory, ring theory, module theory) is absolutely vital (from my limited experience) to progressing further, at least if you want to consider doing stuff like algebraic topology and algebraic geometry – Perturbative Nov 23 '18 at 21:43
  • @Perturbative I'm curious, what did you think of Dummit & Foote? For me it's the worst math book I have ever seen. First, I feel like their proofs sometimes obfuscate things a lot. Second, in other books I usually turn to exercises to solidify my understanding of the material; but in D&F, I felt like many of the exercises were not useful at all; they introduce a lot of stuff that is used later on, but when it's introduced out of context it just seems really random, useless, and not memorable at all. – Ovi Nov 24 '18 at 02:30

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As with many A-or-B questions, the answer is "both".

If you are going to learn both, it is faster to start with the coordinate-free algebra of groups/rings/fields/modules, and then read a linear algebra book.

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