Which one is fundamental to the other between abstract algebra and linear algebra? (soft question)

Question

I saw a post online the other day that mentioned how "linear algebra is one of the most widely developed fields in mathematics in the $21^{st}$ century" and whether it's true or not, I'm not sure, but it did spark a little bit of interest, and so I was wondering:

is abstract algebra fundamental to linear algebra, or is it the other way around, or are the two independent? I'm asking because I'd like to initiate studies into these areas, but am having a little bit of trouble finding out where to begin. If anyone can advise me that would be greatly appreciated.

Also, I'm aware that vectors come up a lot (a lot) in linear algebra, and because of this I was also slightly concerned that a knowledge of multi-variable calculus and its vector spaces is also required. Is it possible for anyone to confirm this?

Question in short: which area of study is more logical to start with: linear algebra or abstract algebra, and why?

Thank you.

PS: It's true, I could've Googled this, but the majority of resources on Google would be objective, and I'm looking for a bit of personal advice, hence I am asking here.

Answer 1

It's more logical to start with linear algebra (perhaps not every aspect in the field, but certainly the fundamentals.)

Understanding how vector spaces and their endomorphism rings work, transformation groups via matrices, and analyses like the Jordan Normal Form, provide vital examples and geometric intuition that can be used in abstract algebra. Understanding fields and vector spaces makes the jump to rings and modules easier, and understanding transformation groups leads you to groups and so on.

It is difficult to learn or make conjectures in an abstract area unless you have a lot of good examples and experience provided by something like linear algebra. I would consider it a bad idea in general to take an abstract algebra course ahead of basic linear algebra and real analysis.

In order to reinforce a foundation for later work with commutative algebra, I would also recommend learning the basics of complex and real analysis. These two things, along with linear algebra, provide an excellent supply of intuition to build upon while learning abstract algebra.

Answer 2

It's normal in every university program I've ever seen to do linear algebra first, so if your goal is self study then the overwhelming evidence is linear algebra should come first.

You certainly don't need any vector calculus to begin learning linear algebra either.