I am in need of a (linear?) algebra book that helps me understand my lectures better. The lectures are entirely proof-based, no applications, very few examples (so few that I often struggle because of the dense notation).
The topics for the second half are (the course is in German, where I am unsure of the direct translation I give the German descriptor):
groups, homogenous spaces, quotient groups, ideals and factor rings, divisors/factorization (?, in German it says "Teilbarkeit"), euklidian rings, integer factorization, modules, free modules, torsion, length of a module, matrices in euklidian rings, Struktursätze für Moduln (= theorems on the structure of modules?), minimal polynom, rational normal form, Smith-normal form, Jordan normal form, äußere Potenz (= Graßmann-Algebra?), bilinear forms, symmetric bilinear forms, eigenvectors and eigenvalues, tensor products, Charakter einer Darstellung (no clue how this translates!), conjugacy classes, Darstellungstheorie (= group representation?), orthogonality and its relations, division into irreducible representations
I believe that this half looks more like an intro to algebra course rather than just a linear algebra course but I believe we do not really dive deeper into the theory. The problem is that a deeper, intuitive understanding is hard to develop. For example, in the first half, we proofed a lot about determinants but the notes never really explained how one should think about determinants. The same pretty much applies to the second half of the course.
What I would like is a book that does not develop mentioned things much further but rather explains how one should think about those things. I cannot see the forest for the trees anymore. I can prove a couple of minute things but how they depend on each other eludes me and I would like to remedy that.
All the best!