# Lang's Linear Algebra: what's next?

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

1. what are the "more advanced" topics in linear algebra that are not covered by Lang's (update: or Roman's) book and where to study them;
2. what are the "modern research topics" in "pure" linear algebra.
• I think this is hard to answer. The snappy response to your last sentence is that there are none, but the more positive viewpoint is that most all modern research uses linear algebra so you had better narrow it down; that's going to be hard to do in this space. I don't think making pure linear algebra the cause of your life is really practical (maybe some people familiar with numerical methods should chime in; I'm sure there's work there). If you think you're drawn to algebraic thinking maybe the next step is to read a book with a broad view of the subject like M. Artin's Algebra. – Hoot Dec 21 '14 at 0:37
• Related math stackexchange questions: What's a good book on advanced linear algebra? and High-level linear algebra book – Dave L. Renfro Jan 23 '15 at 16:59

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.