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.