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What is the inverse of a vector?

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Re: What is the inverse of a vector?

#61

Post by fsloth »

deepsun wrote: post #32
IshKebab wrote: post #28

Earlier quoted context omitted.

I definitely agree but there's no way this is "English/American style". It's because people have grand plans of making their article/book accessible to everyone, and they start off explaining e.g. what a vector is, but pretty soon realise the don't want to write an entire vector algebra textbook so they seamlessly give up and jump straight into Stoke's theorem or whatever. I read a Synopsys simulator manual that expl…

Probably. But why would so many people want to make their article/book accessible to everyone. Let's accept the fact that some topics, like vector algebra, are just not that interesting to everyone.

Vector algebra is at the heart of a fairly large industry - games. I don't think there can be enough of accessible and understandable content from that point of view.

Re: What is the inverse of a vector?

#63

Post by learc83 »

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.

In the US, it’s common to refer to university as school.

Re: What is the inverse of a vector?

#64

Post by ReleaseCandidat »

qayxc wrote: post #54
ReleaseCandidat wrote: post #40

Earlier quoted context omitted.

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 b…

> There are two origins of CS in German universities: electrical engineering and maths.

Ah, I see, thanks for the information.

Which university would be an example for the latter?

Re: What is the inverse of a vector?

#65

Post by captainmuon »

spekcular wrote: post #7

The article begins: >In this post we will re-invent a form of math that is far superior to the one you learned in school. The ideas herein are nothing short of revolutionary. and concludes: > I firmly believe that in 100 years, Geometric Algebra will be the dominant way of introducing students to mathematical physics. In the same way that Newton's notation for Calculus is no longer the dominant one, or that Maxwell's…

I wish I would have been introduced to Geometric Algebra or calculus of forms or whatever it is called during my physics studies. We learned all the conventional things you need for classical mechanics and electromagnetism, like div and curl and BAC-CAB. But there were a couple of things that we were not tought well, which caused problems later. One thing is that at first, a vector was just an N-tuple. But in physics, something is only a vector or a tensor if it behaves under a very specific way during transformations. (The infamous "a tensor is an object that transforms like a tensor"...)

The other thing is that I hit a wall when trying to read theory papers, because nobody ever explained what a wedge operator or a two-form and so on is. I was able to mechanically follow calculations, but never got a real good intuition.

We learned math together with the mathematicians in a very axiomatic way. Without a geometric intuition, I had a hard time understanding why you need dual vectors like one-forms or covariant vectors. They seem just like a convoluted way to write the scalar product.

Re: What is the inverse of a vector?

#66

Post by Grustaf »

nocturnial wrote: post #29
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…

> Ah, another geometric algebra evangelist? Heh, this made me chuckle and this is reason why I read HN comments. I can't help but to picture someone ringing your doorbell early on an sunday and saying: "Hi! Have you considered inverting vectors?" slams door "Who was it?" "It was just another of those damned geometric algebraists"

Haha that reminds me, got to put back that "No beggars or geometric albraists" sign!

Re: What is the inverse of a vector?

#67

Post by mathgenius »

BlackFly wrote: post #5

For those who are interested, this sort of algebra would be known as the [Grassman algebra or the exterior algebra]( https://en.wikipedia.org/wiki/Exterior_algebra ). It becomes much more interesting if you use non-orthonormal bases (or non-euclidean geometry), since then you need to introduce a dual basis and distinguish between contravariant vectors and covariant vectors. When you add derivatives to the mix you end…

Yes, this is exterior algebra. It's also interesting to figure out how this works in ambient dimensions other than three. The author has a table of grades: 0 for scalars, 1 for vectors, 2 for "bivectors", 3 for "trivectors", and they count the number of bases for each of these grades as 1 3 3 1. These basis counts are the dimensions of the (vector space of) scalars, vectors, "bivectors", "trivectors". If you go to two ambient dimensions you get 1 2 1, and if you go to four ambient dimensions you get 1 4 6 4 1. It's Pascal's triangle.

Re: What is the inverse of a vector?

#69

Post by thaumasiotes »

ReleaseCandidat wrote: post #25
thaumasiotes wrote: post #6

Earlier quoted context omitted.

> As vectors, obviously they have inverses, additive inverses. Since vectors don't have multiplication, there is no multiplicative inverse A vector is pretty much by definition also a matrix, and there is a standard way to multiply matrices. You can define several inverses of a vector that way, though you can't define a unique inverse. The standard inner product is of course also an exceptionally typical way to multi…

> A vector is pretty much by definition also a matrix, and there is a standard way to multiply matrices. A standard way to multiply a MxN with a NxK matrix, but none for a 1xN with a 1xN or a Nx1 with a Nx1 matrix - the two possible ways to describe a vector. You have to transpose exactly one of the two vectors. And then you have two possible results, 1xN multiplied with Nx1 yields a scalar (that's actually the 'usua…

Doesn't matter. We already didn't have a unique inverse, but it's perfectly possible to find a left pseudoinverse and a right pseudoinverse, bearing in mind that they're not unique.

Though thinking about it more, it seems like the outer-product-inverse of a vector (a) must be unique if it exists; and (b) is highly unlikely to exist.

> 1xN multiplied with Nx1 yields a scalar (that's actually the 'usual' dot-product/scalar product/whatever you call it)

I'm aware of this, but there are two ways we might conceive of an "inverse":

- Since a vector is a matrix, the inverse of a vector might be defined by matrix multiplication, where A is the inverse of B if AB is "the" identity matrix. This is only strictly defined for square matrices, but the pseudoinverse concept extends it to nonsquare matrices.

- Or, we could go for a more basic sense of "multiplicative inverse", where the concept is that if AB = C, then B = A⁻¹C. This is what I was thinking of when saying that the concept of an inverse doesn't make sense when multiplication is the inner product - if I give you a vector v, and its inner product with some other vector u, there is no way of recovering what u was.

Re: What is the inverse of a vector?

#70

Post by LeanderK »

ReleaseCandidat wrote: post #64
qayxc wrote: post #54

Earlier quoted context omitted.

> 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 b…

> There are two origins of CS in German universities: electrical engineering and maths. Ah, I see, thanks for the information. Which university would be an example for the latter?

Karlsruhe Institute for Technology is an example. Analysis 1+2 and Linear Algebra 1+2 is compulsory for every CS student (the exams are practically identical, just missing a single topic. You can always switch to the math lectures and get a few ECTS more). This is a great background for more advanced, theoretical lectures (CS+Math) but it's lacking electrical engineering lectures and especially there's no electrical engineering practice.
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