I recommend Alan Macdonald's introductory textbook Linear and Geometric Algebra, available from amazon.com. The author calls it sophomore level, and I would agree. This short, inexpensive little book contains a lot of material you'll have seen already, but it's presented from the non-traditional but increasingly popular perspective of geometric algebra. My experience is that geometric algebra provides a simple but amazingly powerful system for unifying many of the ideas of mathematical physics: scalars, vectors, k-vectors; outer (wedge) and inner (contraction) products; complex numbers; quaternionic, Pauli, and Dirac algebras; differential geometry; and much more. The topics of vector calculus and more are treated from a geometric algebra viewpoint in Macdonald's follow-up text, Vector and Geometric Calculus. LAGA and VAGC also introduce the reader to the free computer algebra system SymPy and its geometric algebra module GA.
The same viewpoint is taken in Geometric Algebra for Computer Science, by Dorst, Fontijne, & Mann. Although most of GACS is devoted to computer graphics, the initial chapters give a leisurely and well-written introduction to geometric algebra and its relationship to traditional presentations of linear algebra. GACS shows how projective geometry and conformal geometry can be encoded within the geometric algebra. I have severe reservations about GACS's discussion of the calculus associated with geometric algebra, however; their discussion is not up to the level displayed when discussing the geometric algebra, and I think that many of the calculus formulas displayed are incorrect. Get the revised edition of GACS; the first edition acquired an embarrassingly lengthy list of errata.
Considerably more advanced (graduate level) is Geometric Algebra for Physicists, by Doran and Lasenby. The number of ideas introduced therein is staggering.
For a general course in modern algebra I have two recommendations. Easier to read by far is John B. Fraleigh's A First Course in Abstract Algebra, which has gone through multiple editions. The second recommendation, not easy but still well written, is Algebra (Third Edition, AMS Chelsea Publishing, 1993), by MacLane and Birkhoff (not to be confused with A Survey of Modern Algebra, a classic earlier textbook by the same authors).
Best Answer
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.