Math Books Organized by Area of Mathematics

Actually, if you are a complete novice, Linear Algebra Done Right won't serve you too well. You need to understand basics before approaching it. I should probably include Strang's book as a first introduction, before Linear Algebra Done Right. On a side note, Axler's book is also $120 cheaper than Strang's (which may or may not matter to you). You could also watch the free lectures by Strang online, and then use Axler's book as a supplement. I personally think that Axler's exposition is far more concise and clear.

"For those with more knowledge: in what way is Sheldon Axler's "Linear Algebra Done Right" (quoted in the list above) superior to Strang's "Linear Algebra" for the complete novice?"

A complete novice should go for Strang. Axler says upfront (in the preface iirc it has been a while since I worked through it) his book is intended as the second Linear Algebra book and assumes you have a base in Linear Algebra (matrix manipulations and so forth) already. Axler depends more on a classical "theorems and proofs" approach. Strang doesn't involve proofs etc and is more ocncerned with funamental operations and building intuition and so (imo) is more suited to the complete beginner.

Axler and Strang are really very different books with very different intents. The most striking example is that Gaussian elimination is practically the first thing that Strang covers, but Axler doesn't cover it at all.

I can't speak in general because I've only skimmed Strang, whereas I worked through nearly all of Axler. But if your interest in Linear Algebra is as preparation for quantum mechanics, then Axler is a great choice, perhaps even for a novice (though it does require a bit of mathematical maturity). Axler really hammers you with the idea that a matrix is just a way to represent a linear transformation, and that the numbers in the matrix depend on the basis you choose for the underlying vector space. This way of thinking is very helpful when you learn quantum mechanics.

EDIT: Axler also has the advantage that it is short. Personally, I find it much easier to work independently through short math/physics books than long books.

For real analysis, I'd recommend that you pick up a copy of "baby Rudin" regardless, but for learning real analysis, I'd recommend a book like "A First Course in Mathematical Analysis" by Burkill. Other viable options include "Calculus" by Spivak (it's really a book on real analysis at a level somewhere between a typical freshman calculus book and "baby Rudin"), "Real Mathematical Analysis" by Pugh, or "Yet Another Introduction to Analysis" by Bryant.

They are not distinct areas. However, I feel that dividing them up helps the reader compare apples to apples, rather than having a list of 10 calculus related books.

Precalculus, at least in US high schools, means something like “trigonomety and limits, plus applications”. “Calculus I” means single variable calculus, though the Spivak book they list is much more rigorous than a typical first-year calculus course. “Calculus II/III” in this case means vector calculus, which is enough of its own material to get at least its own semester course. I don’t think I’d quite put Spivak’s Calculus on Manifolds in the same category – maybe more like basic differential geometry/differential topology – but hey, this isn’t my list.

If you look at the divisions being made, its pretty clearly based on how people would look for books. For calculus, lots of people buy books based on classes they take. Hence the division.

No one of any mathematical repute or who took anything beyond calculus would be confused by this so I'm okay with the menu as is.

Another notable book missing from that list is "Topology" by Munkres.

To quote justokay quoting Mike Jordan

http://news.ycombinator.com/item?id=1055389

I've read a number of books from that sequence and they are definitely the core you need to start hitting for advanced stat.