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

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    $\begingroup$ 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 $\endgroup$ – imranfat Nov 7 '16 at 16:16
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    $\begingroup$ 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. $\endgroup$ – RGS Nov 7 '16 at 16:17
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    $\begingroup$ 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. $\endgroup$ – ctst Nov 7 '16 at 16:17
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    $\begingroup$ 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. $\endgroup$ – Mees de Vries Nov 7 '16 at 16:19
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    $\begingroup$ 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. $\endgroup$ – Xam Nov 7 '16 at 16:35
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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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