What is the inverse of a vector? • Hacker News Board

Re: What is the inverse of a vector?

#41

Post by red_trumpet » Wed, Nov 03, 2021, 10:26 AM UTC

oikawa_tooru_ wrote: post #23

Garlef wrote: post #8

While the article is written very nicely, It seems that this is written out of a perspective of some missing knowledge. The basic object that the author seems to be interested in is that of an "algebra over a field" ( https://en.wikipedia.org/wiki/Algebra_over_a_field ). Specifically: Invertability of all elements with respect to the multiplication leads to the notion of division algebra and these have been studied f…

Damn. They don't teach these stuff here atleast not in a computer science curriculum. What degree did you learn? Iss this generally taught in all German engineering courses?

I think also in Germany you have to study pure math to see this. Where I study you even can avoid this by never taking Algebra 1, which is not obligatory (though imo it should be standard. There you learn why polynomials of degree >=5 have no closed form solutions)

Re: What is the inverse of a vector?

#42

Post by LotusFunctor » Wed, Nov 03, 2021, 10:27 AM UTC

Grustaf wrote: post #4

Ah, another geometric algebra evangelist? I can't figure out if GA actually adds anything substantial, or if it merely lets us write some equations in a more succinct fashion. But it certainly looks cool. As to vectors, obviously they have inverses, additive inverses. Since vectors don't have multiplication, there is no multiplicative inverse, but if you define new operations on them, well then that operation can hav…

For me, the substantial thing geometric algebra gave me so far was a newfound appreciation of the seemingly disparate systems: tensors, differential forms, matrix algebra, and also a newfound appreciation of stuff like determinants, conjugate elements in group theory, lie groups and lie algebras, etc., because it helps clarify the relationship between them, and as another user here said, you can get propelled up into some pretty advanced stuff later on (said user mentions the Atiyah-Singer Index Theorem and Hodge theory, but caveat: I've only recently started tacking a crack at the latter. I will say that, OTOH, it's pretty nice to be able to see something like the wiki on Clifford Analysis and realize its familiar territory from geometric calculus).

Re: What is the inverse of a vector?

#43

Post by j7ake » Wed, Nov 03, 2021, 10:29 AM UTC

deepsun wrote: post #24

English/American style of explanation fascinates me. First, they show some algebra formulas and mention dot product and cross product. But then they start introducing a definition of a vector! With images! Why, oh why do you need to waste yours and reader's time to introduce basic definitions, if any reader of the article definitely knows that? If they haven't, they wouldn't be able to read the first paragraph at all…

Do you have an example of "Russian style explanation"?

Re: What is the inverse of a vector?

#44

Post by Sharlin » Wed, Nov 03, 2021, 10:34 AM UTC

Garlef wrote: post #8

While the article is written very nicely, It seems that this is written out of a perspective of some missing knowledge. The basic object that the author seems to be interested in is that of an "algebra over a field" ( https://en.wikipedia.org/wiki/Algebra_over_a_field ). Specifically: Invertability of all elements with respect to the multiplication leads to the notion of division algebra and these have been studied f…

It’s very intentionally written from the perspective of someone who knows linear algebra but maybe not that much, if any, abstract algebra, because that’s what the target audience is. Not university math students or graduates, but highschool graduates, CS/physics/engineering students and graduates, game/graphics programmers and so on.

Re: What is the inverse of a vector?

#45

Post by an1sotropy » Wed, Nov 03, 2021, 10:37 AM UTC

ur-whale wrote: post #15

Geometric algebra (Clifford Algebra) unfortunately came late historically. It's a shame, because the whole theory is a very useful (eg for engineering / applied math) superset of linear algebra. I really wish I had learned this first in my undergrad years, would have made a whole bunch of things way clearer from the get go: differential forms tensor calculus linear algebra etc From zero to geo is a very good video in…

Yes, this. Though Zero to Geo is one of the links at the bottom of the article.

It is really a shame that article does not clarify that, btw, what we've just derived is a re-derivation of a thing that has already been expressed and named, by Clifford, and well-characterized: https://en.wikipedia.org/wiki/Geometric_algebra

Such a bummer to see very slick but very ahistorical articles.

Re: What is the inverse of a vector?

#46

Post by azalemeth » Wed, Nov 03, 2021, 10:41 AM UTC

vanderZwan wrote: post #20

Garlef wrote: post #8

While the article is written very nicely, It seems that this is written out of a perspective of some missing knowledge. The basic object that the author seems to be interested in is that of an "algebra over a field" ( https://en.wikipedia.org/wiki/Algebra_over_a_field ). Specifically: Invertability of all elements with respect to the multiplication leads to the notion of division algebra and these have been studied f…

> It seems that this is written out of a perspective of some missing knowledge. Well, the author talks about what "we learned in school ", not university , so that checks out but only because you two have different audiences in mind.

Indeed. I learnt about vector dot and cross products, basic linear algebra (including diagonalisation, simple Markov chains and similar), partial differentiation, grad div and curl, the volume of a parallel piped and "all that Jazz" in high school, as a 17-18 year old. I learnt about the divergence theorem, Stokes's theorem, multivariate integration, integrating factors and higher order ODEs and simple PDEs, Fourier transforms and other integral transformations and similar in the first year of university (studying Physics).

This is not uncommon in the UK – but it is also not common either, and depends on exactly what A-level modules you did. My understanding is that it's quite rare to do exactly this in high school in the united states – but there, I think limits are taught much more heavily. I think having a clear, short statement of having "assumed knowledge" somewhere probably helps avoid these issues.

(I thought the article was excellent, and beautifully illustrated!)

Re: What is the inverse of a vector?

#47

Post by SantiagoQ » Wed, Nov 03, 2021, 10:44 AM UTC

Semantically, the inverse of a vector is something that has no magnitude nor direction.

I wonder what would that look like (mathematically), and what surfaces or fields would it create?

Re: What is the inverse of a vector?

#48

Post by Grustaf » Wed, Nov 03, 2021, 10:45 AM UTC

deepsun wrote: post #24

English/American style of explanation fascinates me. First, they show some algebra formulas and mention dot product and cross product. But then they start introducing a definition of a vector! With images! Why, oh why do you need to waste yours and reader's time to introduce basic definitions, if any reader of the article definitely knows that? If they haven't, they wouldn't be able to read the first paragraph at all…

There's an urban legend at Swedish universities that American text book authors get paid by the word, and that's why their books are so incredibly verbose.

Re: What is the inverse of a vector?

#49

Post by jefftk » Wed, Nov 03, 2021, 10:46 AM UTC

> The similarities are so striking that we might think of them as "pseudovpseudovectors". But I won't write them this way because I think that obscures their true nature. Written this way it looks like a bivector only encapsulates three degrees of freedom!

> Instead, I will use: ... Because it forces us to remember what those coefficients are attached to. Knowing that a bivector contains five degrees of freedom, can you figure out what the other two describe?

I'm confused here and don't understand why they keep saying a bivector has five degrees of freedom. If you can uniquely identify one with three scalar coefficients, doesn't it only have three degrees of freedom?

Re: What is the inverse of a vector?

#50

Post by codeflo » Wed, Nov 03, 2021, 10:49 AM UTC

The writing is cute and the animations are nice, but none of it makes any sense. I stopped reading at

> It is important to remember that bivectors have a certain redundancy built into them in the sense that s a ⃗ ∧ b ⃗ = a ⃗ ∧ s b ⃗ s a ∧ b = a ∧s b . We can write them using 6 numbers or 3 numbers, but they actually convey 5 degrees of freedom.

Three (real) numbers have three degrees of freedom, by definition. (And nothing about complex numbers was mentioned.) Is this a parody I don’t get? I feel like I have wasted ten minutes on nonsense.