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What is the inverse of a vector?

mattferraro.dev

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Re: What is the inverse of a vector?

#51

Post by Garlef »

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?

> What degree did you learn?

Mathematics (I started a PhD but left academia in favour of starting a company). My focus was on category theory / algebra.

Re: What is the inverse of a vector?

#53

Post by tgb »

codeflo wrote: post #50

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 n…

You're right about that being wrong, and the author makes the same mistake consistently, but otherwise it looks correct. Some steps have details elided where it maybe should have been noted that things were being skipped, but with correct results. I think it's wonderfully written and a great exposition.

Re: What is the inverse of a vector?

#54

Post by qayxc »

ReleaseCandidat wrote: post #40
oikawa_tooru_ wrote: post #23

Earlier quoted context omitted.

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?

He is talking about math studies. Not much maths in engineering/CS in germany either. But nobody is holding you back from visiting 'real' math lectures ;)

> Not much maths in engineering/CS in germany either.

That's simply not true. It depends entirely on the particular institution and its roots.

There are two origins of CS in German universities: electrical engineering and maths. At universities where CS originated as a subfield of maths, undergrad CS education is very similar to a maths undergrad to the point that most of the tests/mid-terms are basically identical between CS and maths.

If on the other hand CS came from the electrical engineering department, the focus is significantly less on maths and the lectures are very different indeed.

So you'd have to look at the history of each university and where the CS department originated to find out.

Re: What is the inverse of a vector?

#55

Post by conradludgate »

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?

Speaking from personal experience, not all of us on HN have computer science degrees.

I studied mathematics in the UK and can confirm I learnt a lot about fields and other forms of linear and abstract algebras

Re: What is the inverse of a vector?

#56

Post by puzzledobserver »

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…

(I have only briefly skimmed the article, but ...)

Material such as this serves to remind the reader of what they already know, and contextualize it in a way that is relevant to this article. The article begins by telling the reader, "We all know what scalars and vectors are---here they are---but have you wondered what if ...", and taking the reader beyond. The introduction which you seem to find objectionable is only a small part of a much longer article.

In addition, different readers have different, mildly different notation styles. These introductory blobs inform the reader of the language in the article, and are essentially a friendly statement of definitions.

A third purpose is rhetorical: readers sometimes get stuck while reading text, and these parts of the article work as anchor points where they can loop back and "synchronize" with the writer.

Re: What is the inverse of a vector?

#58

Post by tgb »

jefftk wrote: post #49

> 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…

Yes, only three. As defined, two bivectors are equal if their areas are equal and if their oriented planes are equal. Therefore two more degrees of freedom are absorbed by taking rotations of the two vectors in the plane.Along with the rescaling the author noted, we're down to three from six.

Re: What is the inverse of a vector?

#59

Post by codeflo »

tgb wrote: post #53
codeflo wrote: post #50

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 n…

You're right about that being wrong, and the author makes the same mistake consistently, but otherwise it looks correct. Some steps have details elided where it maybe should have been noted that things were being skipped, but with correct results. I think it's wonderfully written and a great exposition.

Thanks, that helps. When I notice errors in the stuff I already know about, it find it hard to trust the other information that’s new to me.

Re: What is the inverse of a vector?

#60

Post by [deleted] »

azalemeth wrote: post #46
vanderZwan wrote: post #20

Earlier quoted context omitted.

> 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…

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