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.
The secret is that math is too big to fit into any one book. It takes many books with many perspectives you learn the generality and applicability of the topic. The average person who couldn't learn a topic read one book on the topic. The average mathematician read 5 or 10.
Good point. Even in maths, "bias" is inevitable. Every mathematician has a point of view. There's no objectively best curriculum or way of doing things, understanding things, solving problems etc.
(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.)
Thanks. I found LADR not that helpful - I use linear algebra constantly but only on a very elementary level. After reading your message I now feel it's ok not to find that book applicable to my pursuits and that I'm not missing some concept I failed to grasp from the book.
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.