> What makes LADR superior to this book?
Axler's approach for determinants.
Axler's approach for determinants.
Axler defines determinant as (up to a sign) the constant term of the characteristic polynomial, and he needs two different definitions for characteristic polynomial, one over R and one over C. Now what if the ground field is something else? Do we need yet another definition of characteristic polynomial in order to define the determinant? What if you are doing linear algebra over a commutative ring?
LADR actually presents a very narrow view about linear algebra : it treats linear algebra merely as finite-dimensional functional analysis. The readers can be hit hard when they need to do other (computational or theoretical) stuffs. Similar concerns had been voiced on the internet before. In particular, I think Darij Grinberg's comments (below the answer https://mathoverflow.net/a/16996) on LADR are rather spot on.
It's fine if you find LADR helpful. The book does have its merits (I like its clear and fluent writing and its neat proofs), but it has also its own shares of problems and there are other nice choices of books in the wild.
(I'm more familiar with this phenomenon in philosophy, where the greater the philosopher, the more they have entirely their own way of looking at things, untranslatable into another tongue, which you just have to come to understand on its own terms. A summary of their views leaves out the personal aspect, the style, the way of thinking, and will seem dead.)
Say what?
This is probably the correct generalization of the volume definition.
You don't start kids with complex numbers until they can handle reals. You don't even start negative numbers until they can handle positive numbers.
The only beneficial traditional treatment I can think of is saying "determinant is volume", and that's what Axler does.
From there, one can look into alternating forms, convince oneself that an alternating n-linear form does the same thing, and obtain a formula for it (e.g. summing up signed products of numbers on the diagonals over all permutations of columns).
If by "traditional" you mean: "Here's an insanely complicated formula that does something magical. Learn to compute it. On page 5, we'll prove that it tells something about independence. Oh, and we'll mention volume on page 10 briefly" -- then I not only think this is not useful, I think it's outright harmful.