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I've completed the study of Lang's Linear Algebra ($3^\text{rd}$ edition). To put it simply, I have enjoyed the subject and I would like to know "what's next".
In other words, I would like to know
I would go for some later chapters of Advanced Linear Algebra, from Steven Roman, and Linear Algebra and Geometry, from Kostrikin and Manin. About open problems in Linear Algebra, you can take a look at the comments in this question: Are there open problems in Linear Algebra?. Particularly, I find it difficult to find open problems in linear algebra, since (in my point of view) a great part of this is mainly language for more advanced topics such as functional analysis, differential geometry, etc.
I'd recommend learning about other topics in mathematics that strongly employ the language and techniques of linear algebra. The two clearest examples to me are functional analysis and differential geometry. In functional analysis you get to see the ideas of vector spaces, except these are expanded to the cases of infinite-dimensional Banach and Hilbert spaces, the latter of which is fundamental to the structure of quantum mechanics. In differential geometry linear algebra is needed to understand tangent spaces and the derivatives of multidimensional maps between manifolds, etc.
As others have stated above, it's probably difficult to find modern "pure" linear algebra research, and the "more advanced" topics you're looking for might be found in other areas of mathematics that strongly employ linear algebra. You may also want to learn about representation theory as well.
Let me concisely but properly address your questions.
1) Learn about tensor products and multilinear algebra in general. (Since you mention that you read Lang's linear algebra book, there is a more advanced algebra book by Lang, called - guess what - Algebra; it is a standard reference.) These things are totally crucial in e.g. commutative algebra, algebraic geometry, algebraic topology, mathematical physics, etc. They also allow you to give more canonical definitions and viewpoints on classical topics such as the determinant or the trace. Lang's treatment uses modules (over commutative rings), which are the natural generalization of vector spaces, and I strongly recommend you to learn about module theory (also from Lang's Algebra). Then the natural way to proceed is to learn about homological algebra (Lang's Algebra contains an introduction, but there is a lot more to learn). (A professor of mine called homological algebra "smart linear algebra", which is in some sense true.)
2) I actually asked this question myself; a professor of mine responded by saying that "linear algebra is trivial and not a subject of current research", so I can only leave you with this. Note that this is what he says and not what I think; personally I think that there are probably not a lot of people who consider themselves "pure linear algebraists" (in the same way that probably hardly any professional mathematician would described his field of research as "calculus" in the narrow sense), but quite a few who study numerical linear algebra.
I really like Introduction to Matrix Analysis by Richard Bellman.
Have you studied abstract algebra? I believe it is a natural step after learning linear algebra, as a vector space is often one of the (if not the) first abstract algebraic structure a student encounters. Perhaps take a look at Dummit and Foote's text. It also covers much of what an "advanced" linear algebra text might, and much more.
As someone mentioned above, your question is really difficult to answer, especially the modern topics in "pure" linear algebra. It is hard to say whether some topics are "pure". Anyway, I think Brian C. Hall's book (maybe GTM222) about matrix lie group, somehow may help you.
