"Don’t think of them as rotations, think of them as a unit vectors of a new coordinate systems."
This makes it clear to me that many people have been very poorly served by their linear algebra courses if that is not obvious.
This makes it clear to me that many people have been very poorly served by their linear algebra courses if that is not obvious.
Years later when I got into graphics programming I found myself dealing with linear algebra concepts but this time it made much more sense to me. Being able to apply the ideas to real problems with visual solutions made all the difference.
Ohhhhh, now I get determinants. I just thought they were some easy way to classify a matrix, not that they had any special meaning. Not that they were also all the eigenvalues multiplied too.
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.
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.
I agree with you, that in practical paper computations (e.g. on exams) that determinants are an indespensible practical tool.
E.g. the determinants of the rotation matrices in the article are all 1, because they don't scale or reflect anything, just rotate.
I think the important detail is if you're changing from right-handed bases to left-handed or vice versa during the transformation.
I fail to see the point you tried to make. Both transformations preserve volume. Why did you expected volume to change if you rotate something?
I don't know where you thought I expected volume to change if I rotate something.
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.
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.
https://www.amazon.com/Vector-Calculus-Linear-Algebra-Differ...
math majors would probably supplement this with additional material
On the other hand though, wouldn't you lose the generality? Part of my undergraduate geometry and linear algebra course was on fields, groups and rings and that was built in order to show that once you can define a certain set of operations on a mathematical structure you can do "math" on such structures "as if" they were regular numbers (limitations apply).
Think of matrix fields (not sure if it would be the correct translation, in Italian they were called "campi matriciali").
That was kinda mind blowing to me but at the same time it made perfect sense because of the generic definitions underlying.
Not sure how you could go ahead and explain that with regular 3d-world physics.
Why does it matter that it's unit vectors in a new coordinate system? I don't really care about the new coordinate system, I care about rotating something to this position in my old coordinate system. Saying "oh yeah it's <1,1,1> in this coordinate system" doesn't really help because I still need to translate <1,1,1> in the old coordinate system to <1,1,1> in the new coordinate system... and now I've got the same problem again.
I mean, I get that a rotation matrix, or any matrix for that matter, represents unit vectors in a new coordinate system. If I got nothing else from that course I at least learned that a matrix represents like 50 different concepts like systems of equations, coordinate spaces, etc. A big part of my course was applying rules for one representation of a problem to another equivalent representation to a problem. What I never understood is why it matters how you're representing the problem. Identifying that a rotation matrix is just unit vectors in a new coordinate space doesn't help me solve my problem of wanting to rotate something, yet it felt like 75% of my course was about converting problems from one representation to another.
In 3d graphics (or physics or what have you) you're constantly moving between coordinate systems/frames of reference because certain things are very easy to represent in a local coordinate system but are difficult to work with in a global one
The question of "how to rotate something" rests on operational semantics. The question of "how is this related to that" rests on denotational semantics, and is much more general. The beauty of linear algebra is how closely the two are related: every linear transformation (the relationship) gives a matrix given a choice of basis, and matrices have a direct algorithm for transforming coordinate vectors.
That's exactly what all square non-singular matrices with unit column spaces are. In fact, all square non-singular matrices are coordinate transformations. That's pretty much the central point of linear algebra.
Framing rotation matrices as changes to the coordinate system doesn't add much info like framing polynomials as continuous functions. Technically it's true, but it doesn't add much to clarify.
SNES programming also directly introduces people into the notion of 2d vectors advancing over an original axis, which ends up in a rotation in mode 7 stuff.
For instance, I have a broad math background, nothing too deep, but enough to have an intuition on how complex things worked. One of the defining projects of my career to date was an embedded signal processing project given to me because "it seems like you know math" -- and I figured it out on the job by intuiting my way through. I figured out Fourier analysis and windows and FIR filters by intuition, and was able to use the pieces I had available as API to do something truly impactful.
Most people won't need most things from school, but if you're the one who knows that one thing that's needed that one time, it can be disproportionately rewarding.