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Which one should I really concentrate on learning first? I want to truly understand both of them. (I'm interested in mathematics for mathematics's sake)

Can you recommend any good introductory textbooks on both of them?

I've gotten through the first couple of chapters in my linear algebra book, but what comes after eigenvalues and eigenvectors? For some odd reason, this seems to be the stopping point for most introductory linear algebra books.

After I get down the concepts of linear algebra, I want to re-learn it, only in a less intuitive sense, but in a more abstract algebra-ish way. I would prefer them to be defined using algebraic structures instead of with applications. However, I only want to learn it like this

*after*I understand it through an intuitive sense.

Thank you for your help!