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.
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.