Hacker News Reader

Comment by unao | Hacker News Reader

The neglect of the exterior algebra is the mathematical tragedy of our century.

Why has it been neglected?

If you look at the paper you will see that it uses a relatively advanced mathematical language. Did you understand the significance and meaning of exterior and geometric algebra from that? If not, then there you have a partial explanation for it. It is much easier to just understand vectors and matrices and maybe tensors for practical and applied work.

I do not agree with this. The simplest mathematical concepts can look very complicated when presented in a formal way.

Learning the geometric product in high school wouldn't be more difficult than learning the dot and cross products, and would make obvious difficult to grasp concepts as complex numbers and even quaternions.

There are historical reasons for which we do not learn this from another point of view, and in my opinion it is, indeed, a great tragedy. A tragedy that I hope will be remedied some day.

Disclaimer: I deal everyday with 3D rotations. Euler angles have been traditionally used in my field, but they present many problems. Everybody knows we could do better with quaternions, but very few people understand them. I have shown many people how to interpret what quaternions are from geometric algebra concepts and I have not yet found anybody who doesn't think it is much more approachable that way.

I do not disagree with you! For example the book http://faculty.luther.edu/~macdonal/laga/ gives an elementary introduction to the subject.

I have learnt about geometric algebra just this year, and applied it to compute the graphics in an app I wrote for a customer. It was a real eye opener, concepts that I struggled with before were really simplified by using geometric algebra.

BUT: I would never have guessed the usefulness of it for me from this paper.

Second recommendation for laga.

The formalism is pretty hard to grasp indeed, but I guess it concerns any mathematical theory. From what I know Exterior / Geometric Algebra is much simpler and more intuitive than let say Linear Algebra.

That's wrong, as Geometric Algebra IS linear algebra. So you first have to understand normal linear algebra (vector spaces, inner product, etc.) and then you can properly understand geometric algebra.

You have it backwards. To understand the inner product you first need to understand the geometric product. ;-)

More seriously though, “linear algebra” is often used to mean “matrix algebra”, which you do not need to understand the basic concepts of geometric algebra. Coordinate-free concepts in linear algebra are geometric algebra concepts, and can be easily taught in a first course on the subject.

What you do need to do is first learn about Euclidean vectors as displacements of Euclidean points (and have some basic grounding in Euclidean geometry of points and lines and circles), after which you can learn about the geometric product of vectors, and the various kinds of multivectors and derived products (e.g. the inner product) which are produced out of that product.

Students can wait until after they have studied the basic concepts to learn more generically about quadratic forms, arbitrary linear transformations (which can be extended to multivector transformations via the “outermorphism”), and so on. And might never need to get into mathematicians’ more abstract/formal concepts of rings and modules and Lie groups and so on, though if they do want to they’ll have some better examples and better intuition about it.

Hmm, that is interesting. Let me cite a book Geometric Algebra for Computer Science:

To understand the structure of the book, you need a better feeling for what geometric algebra is, and how it relates to more classical techniques such as linear algebra.

I followed first chapters of this book and it does not require any knowledge of Linear Algebra.

I read that book as well, it is great for motivation, but mathematically a little sloppy (otherwise you wouldn't come away from it thinking you don't need linear algebra...). It doesn't say so, but it first introduces the linear algebra you need to know: vector spaces, inner products, etc., but without rigorous proofs for basics.

I'm not sure if I would say LA must be learned first, certainly a study of vector spaces and inner product spaces, but not necessarily a complete course in LA. But you will understand each better by understanding the other. If you understand neither, this book covers them together. http://faculty.luther.edu/~macdonal/laga/

I think there is a lot of misunderstanding out there what LA is. For example, LA certainly does not have to include the cross product. Just do introduce the basics of LA like vector spaces, linear mappings, bases, eigenvalues, singular value decomposition, etc. can easily take a semester to grasp properly (if you haven't had any exposure to that sort of thing before).

might seem funny but multilinear algebra (ie geometric algebra) is more intuitive to me than linear algebra (ie vector algebra/calc..)

I don't know what the quotation author meant. Exterior algebra has definitely been well understood and frequently used tool in 20th century.

You are most likely right, but it is still kind of a niche. If you take a look at curriculum of Computer Science studies it is pretty non-present. From what I know Exterior / Geometric Algebra is superior in many ways to Linear Algebra: first it is backwards compatible and second (apparently) it is much more intuitive. But still Linear Algebra is pretty standard.

Exterior algebra is not particularly useful in Computer Science, and that's probably it's non-present. There are other things competing for place in curriculum, and they probably deserve it more.

Exterior algebra is not any more superior to linear algebra than multiplication is superior to addition. Both are important, there are important connections between the two, and you definitely need to understand addition first before you understand multiplication.

Reply on news.ycombinator.com