| Nobody teaches the same topic in the exactly the same way. There is, for one, far more material in any given subject than an introductory course or textbook can (or should) cover, so the author/instructor must choose what to include. Plus, the order of presentation matters. For example, here are two standard ways of introducing the real numbers: #1 (Dedekind cut). Picture a square of side length 1. The length of the diagonal, √2, cannot be represented by a ratio of integers, so we need a new number system to represent it. These numbers "in between" rational numbers are called irrational numbers, and together they form the real numbers. #2 (Cauchy completion). Non-repeating decimals, such as π ≈ 3.141592, cannot be represented by a ratio of integers. We call such decimal numbers irrational numbers. Any number representable by a (finite or infinite) decimal is called a real number. You can deduce #2 from #1, and vice versa. It's entirely up to the author/instructor to decide which one to start with. Lastly, there is always a better way to explain the same material. |
1) Gilbert Strang's "Introduction to Linear Algebra" was great because Gilbert goes straight to intuitions, the proofs are simple, most exercises have answers, but it does not cover advanced material. I used this book for self-teaching. You could probably learn from it with just high-school level maths. Good for engineers.
2) Hoffman and Kunze's "Linear Algebra" was given as a textbook for my first LA course. While it covered some topics that weren't found in many other textbooks and are not really "standard curricula" in many other universities for (jordan normal form, rational canonical form). I found it more similar to a reference than a textbook; it is intended for math majors. The proofs are imho a bit obtuse and it usually introduces topics without much justification. Determinants are introduced early.
3) Axler's "Linear Algebra Done Right" OTOH covered many of the topics in Hoffman&Kunze but the organization and the proofs were (imho) mucho more clear and motivated. Also intended for math majors. No determinants until the end.