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What is the inverse of a vector?

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Re: What is the inverse of a vector?

#91

Post by civilized »

fouronnes3 wrote: post #84
civilized wrote: post #82

Earlier quoted context omitted.

Ironically, this post is an abuse of the concept of "weak typing". There's nothing "weakly typed" about, say, the plus sign being used to add both numbers and sets, or a dot being used for both multiplication of numbers and the dot product of vectors. It just means those symbols dispatch based on the types of their arguments, which is perfectly consistent with strong typing (cf. the Julia language). The situation tha…

That's 100% fair, but while I could have made my point using more rigorous language, I think it's still valid. I'm not talking about multiple dispatch based on type which is fine, I'm talking about actual abuse like using fractions to mean derivatives, omiting non obvious parameters, confusing function and value of function at a point, etc. I could go on. Physics notation is an even worse offender btw. Take for examp…

That's interesting. I'm not quite sure what I would call those issues, but I agree they can be tricky.

Many of those abuses are for a very good cause. Leibniz notation is very powerful, for example, but it's hard to master and physicists really go nuts with it.

For E[X|Y], all you have to remember is that conditional expectation yields a function f(Y) of the thing you conditioned on, and f(Y) is itself a random variable that you can take the expectation of. This property is canonical and baked into the formal definitions. It's not an abuse.

However, I do fault some machine learning types a bit for abusing probability and statistical notation. For example, the Elements of Statistical Learning book extensively overloads E[], P(), and other symbols and operators in ways that it doesn't even bother to define. They randomly throw subscripts and decorations onto all sorts of symbols and don't even bother to tell you if those decorations mean they're marginalizing, conditioning, or something else. The book has no glossary of symbols and operators and no preliminary chapter setting out notation, which is unusual for such an enormous book full of hundreds of equations. It would be impossible because the book is a hodge-podge of symbols that change from paragraph to paragraph.

Re: What is the inverse of a vector?

#92

Post by Jensson »

fouronnes3 wrote: post #84
civilized wrote: post #82

Earlier quoted context omitted.

Ironically, this post is an abuse of the concept of "weak typing". There's nothing "weakly typed" about, say, the plus sign being used to add both numbers and sets, or a dot being used for both multiplication of numbers and the dot product of vectors. It just means those symbols dispatch based on the types of their arguments, which is perfectly consistent with strong typing (cf. the Julia language). The situation tha…

That's 100% fair, but while I could have made my point using more rigorous language, I think it's still valid. I'm not talking about multiple dispatch based on type which is fine, I'm talking about actual abuse like using fractions to mean derivatives, omiting non obvious parameters, confusing function and value of function at a point, etc. I could go on. Physics notation is an even worse offender btw. Take for examp…

That is statistics, not pure maths. But yeah, statisticians abuse a lot of notations like that, just like physicists. I haven't seen pure mathematicians make such unclear notation.

(Statistics is as much maths as theoretical physics is, both are technically mathematics but in practice the field is handled in a very different manner since they are applied and intended to solve a specific set of real world problems and hence not pure)

Re: What is the inverse of a vector?

#93

Post by yobbo »

fouronnes3 wrote: post #77

As a programmer it seems to me that the number one problem of math notation is that it's weakly typed. There's abuse and reuse of notation everywhere, which makes learning it needlessly difficult. I want a strongly typed fork of math notation. 90% of existing math notation would just be laughed at if it had to go through code review.

Yes, one usually needs to "intuit" the meaning when there are mixes of subscripts/superscripts/parameters, assumed definitions, mixes of variable naming conventions, the order of function arguments randomly adjusted to taste, invented symbols and syntaxes.

Since no machine is ever going to read it, it's all up to individual taste and prejudice.

How would a math linter work? Don't see it, but this is one of the things that makes maths a different world from programming.

Re: What is the inverse of a vector?

#94

Post by da39a3ee »

> A scalar is a point on a number line.

This is going to confuse readers. A point on a number line is a 1D vector; in other words it is a unit vector pointing along that number line, multiplied by something which scales its length. It’s the latter dimensionless and directionless quantity that’s the scalar.

Re: What is the inverse of a vector?

#95

Post by timerol »

Sharlin wrote: post #44
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.

Yeah. It's written to teach people who do not have (what some might consider) the standard prerequisite knowledge. I have no doubt the author understands algebra over a field. The author recommends teaching Geometric Algebra as a pedagogy at the end

Re: What is the inverse of a vector?

#96

Post by matt-noonan »

yodon wrote: post #87
matt-noonan wrote: post #79

Earlier quoted context omitted.

No, this is wrong. Geometric algebras aren't division algebras in general: they usually have zero divisors. Objects that live in a single grade are invertible, but composite objects don't always have multiplicative inverses. As a concrete example, consider the elements 1 + x and 1 - x. Their product is 1 + x - x - xx = 1 + x - x - 1 = 0. So certainly 1 + x doesn't have an inverse, either.

Presumably you also meant to say "when X^2 = 1"

Yes, I meant that equation to be interpreted in the GA used in the article. But essentially all geometric algebras also have zero divisors, for similar reasons.

Re: What is the inverse of a vector?

#97

Post by LotusFunctor »

Cybiote wrote: post #86
chriswarbo wrote: post #36

Earlier quoted context omitted.

> Geometric algebra is, as the article points out, a more powerful version of the usual vector notation > the stuff that's already taught is better These two statements seem contradictory. > But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free) The author made no claims about tensor notation or differential forms; perhaps t…

spekcular is correct. One could draw the following graph very roughly capturing how these concepts developed: Grassman’s exterior algebra -> Differential forms Cartan’s Exterior Derivatives -> Differential forms Quaternions -> Clifford Algebra exterior algebra -> Clifford Algebra -> geometric algebra Differential forms are a particular kind of tensor and tensors can be defined in terms of multilinear maps. As spekcul…

From my view, it goes both ways: geometric algebra/calculus is a more transparent version of the standard approach and the translation back to it is also a relatively small delta to pick up.

Either way of going about what is in essence the same material entails becoming familiar with multivectors, the wedge product, and multilinear algebra, whether you do it through geometric algebra or the standard approach.

Re: What is the inverse of a vector?

#98

Post by Grustaf »

fouronnes3 wrote: post #77

As a programmer it seems to me that the number one problem of math notation is that it's weakly typed. There's abuse and reuse of notation everywhere, which makes learning it needlessly difficult. I want a strongly typed fork of math notation. 90% of existing math notation would just be laughed at if it had to go through code review.

That's a very common criticism, but I don't think mathematics would work if you insisted on being 100% explicit all the time.

Clear and short notation, that is just unambiguous enough, is a very important factor, without it books wouldn't just be much longer, I'm not sure we'd even be able to understand it.

Re: What is the inverse of a vector?

#99

Post by kevinwang »

tgb wrote: post #58
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.

Interesting, I was confused about the same thing. So the author is not correct when they say a bivector has 5 degrees of freedom?

Re: What is the inverse of a vector?

#100

Post by LotusFunctor »

mathgenius wrote: post #67
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 tw…

Pascal's triangle, and also with a transparently power-set flavor to it :)
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