I follow Derek Holt's lecture notes and my head only preparing the lectures. But this is what I can advise on books.

The best book covering all second year Algebra (Algebra-1 and Algebra-2 together),
which I recommend and which is worth buying, is

"Algebra" by Michael Artin.

Otherwise, try to browse the library shelves around QA261 and if you find a reasonably looking book (it must contain Jordan Normal Forms and Quadratic Forms), bring it in, and I will tell you how useful this will be. In general, most of linear algebra textbooks cover parts 1-3. You can find them in many styles. One of the best 

"Russian" style textbooks is

"Foundations of linear algebra" by Malcev,

"British" style

"Finite-dimensional vector spaces" by Halmos,

"US college" style

"Linear algebra and its applications" by Strang

and "French" style

"Elements of mathematics. Algebra, Part I: Chapters 1-3." by Bourbaki. (:-)

For part 4, you would need "higher algebra" or "abstract algebra" textbooks, they are located around QA251 in the library. If a book contains "modules over principal ideal domains", it covers our part 4 but from somewhat higher point of view. For example, Artin or Bourbaki will cover it.