That's one concrete way to view determinants, but it's not _the_ way to think about them.
Determinants are kind of abstract and even inelegant. This is because it's defined in terms of coordinates rather than abstractly. A coordinate free definition exists, but it's very complicated.
Ideally they would not be taught at all, but it's useful for doing exam problems. Cramer's rule in particular is quite useful when solving simultaneous equations in engineering under exam conditions, for example. Even in computer/engineering applications, row reduction/diagonal normal form type are used instead.
Determinants are a way to get a scalar number out of a matrix. You have linear forms, bilinear forms, and multilinear forms, the latter of which determinants fall under.
There are certain properties you would like the determinant to satisfy, such as det(AB) = det(A) det(B), so that the determinant can give you an idea of whether a matrix is invertible. Eventually you work out that a determinant _has to be_ a multilinear form with certain properties (alternating, skew symmetric).
Again this is all very complex and messy. Initially 'linear algebra' was just the study of determinants, because people thought of math in terms of solving equations. Linear algebra, as it's known today, came later.
Bilinear forms turn up all the time, so ideally you would first learn bilinear forms, then learn the properties a bilinear form can have (e.g. alternating, skew symmetry, etc), and then learn determinants in that context.
It's a respectable idea that determinants should not be as important as they are on a pedagogical point of view imo. But they can be extremely useful.
From my point of view, matrices are a bad abstraction in general.
For example if we have a linear or a bilinear form, both can be written as matrices of the same size, we can no more differentiate the two objects.
The trouble comes when we want to apply some transformation over those matrices.. as they do not use the same transformations laws..
Namely A^-1.L.A and A^T.B.A
This is a very usual error in computer graphics when we want to make some transformation on a 4x4 matrix and we do not know which kind of object it is.
Halmos are always good books. I didn't know he had this viewpoint on determinants though.
What I meant is that on a pure mathematical point of view they are seen as inelegant as they require a basis to be defined. I don't have a problem with that when you are doing maths on manifolds etc..
From a practical perspective, a lot of engineering problems go from 1 to 4 dimensions and are basis dependents. And the determinant becomes a useful tool.
He doesn't have a particular view on determinants, I got my views on determinants more from Axler than from Halmos. But Halmos' book is linear algebra as preparation for functional analysis, so he tends not to choose a basis for proofs.
I agree with you, that in practical paper computations (e.g. on exams) that determinants are an indespensible practical tool.
Determinants aren't inherently abstract, they're quite practical, especially in the article's context: they tell you the scaling factor of the transformation the matrix represents, and whether it does a reflection.
E.g. the determinants of the rotation matrices in the article are all 1, because they don't scale or reflect anything, just rotate.
The matrix {{1,0,0},{0,-1,0},{0,0,-1}} has a determinant of 1 too, but it reflects along two different axes. So there's no way to distinguish from the determinant between the identity and between a reflection across two axes.
I think the important detail is if you're changing from right-handed bases to left-handed or vice versa during the transformation.
Two reflections make a rotation. Your matrix is a rotation matrix that does no reflection (everything will be oriented the same way after being transformed).
> Determinants are kind of abstract and even inelegant.
No they really aren't. Determinants represent the change in volume in a transformation.
> Ideally [determinants] would not be taught at all, but it's useful for doing exam problems.
This is outright ignorant and wrong. Determinants are fundamental to a myriad of central concepts in calculus and also in engineering fields. For instance, volume integrals or any integral over a parametrization rely on determinants.
I did degrees in both math and engineering, please give me the benefit of the doubt that I'm not making a 1st year university level mistake.
In the specific examples you give, 2-forms could be introduced instead to yield the final formula for the Jacobian. I don't think this is an overkill, because it's possible to use 2-forms without precisely proving the theorems etc about them. This is more convenient for math/physics people, and engineers care more about the final formula than about how exactly it was derived.
As another comment here pointed to, the idea that linear algebra is better off without determinants was popularized by Axler - I'm not making this opinion up out of nowhere.
Determinants are kind of abstract and even inelegant. This is because it's defined in terms of coordinates rather than abstractly. A coordinate free definition exists, but it's very complicated.
Ideally they would not be taught at all, but it's useful for doing exam problems. Cramer's rule in particular is quite useful when solving simultaneous equations in engineering under exam conditions, for example. Even in computer/engineering applications, row reduction/diagonal normal form type are used instead.
Determinants are a way to get a scalar number out of a matrix. You have linear forms, bilinear forms, and multilinear forms, the latter of which determinants fall under.
There are certain properties you would like the determinant to satisfy, such as det(AB) = det(A) det(B), so that the determinant can give you an idea of whether a matrix is invertible. Eventually you work out that a determinant _has to be_ a multilinear form with certain properties (alternating, skew symmetric).
Again this is all very complex and messy. Initially 'linear algebra' was just the study of determinants, because people thought of math in terms of solving equations. Linear algebra, as it's known today, came later.
Bilinear forms turn up all the time, so ideally you would first learn bilinear forms, then learn the properties a bilinear form can have (e.g. alternating, skew symmetry, etc), and then learn determinants in that context.