What is the inverse of a vector?
mattferraro.dev
What is the inverse of a vector?
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Re: What is the inverse of a vector?
#2love your writing, can anyone recommend blogs like these
Re: What is the inverse of a vector?
#3I am curious, how could one apply these insights to inference over sets of vectors generated by creating embeddings over things like photos etc? I understand well the ideas of +/- for things like word2vec, but what would multiplication and inverse mean in this context?
Re: What is the inverse of a vector?
#4Ah, 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 have an inverse but that is not really "the inverse of a vector" anymore.
Re: What is the inverse of a vector?
#5For 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 up in differential geometry.
Re: What is the inverse of a vector?
#6Ah, 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…
> 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 multiply vectors, but the concept of an inverse there doesn't make much sense.
Re: What is the inverse of a vector?
#7The 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 actual equations for Electromagnetism have been replaced by Heaviside's, textbooks will change because a better system has come along.
These claims are wrong.
There are three standard notation methods in physics: vectors, tensors, and differential forms. Geometric algebra is, as the article points out, a more powerful version of the usual vector notation. 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). [I'm oversimplifying a bit, but a full discussion is too long for a comment here.]
Anyway, geometric algebra is not some esoteric secret. People know about it and have decided not to teach it, because the stuff that's already taught is better.
[I picked up this specific phrasing from another user here, knzhou, which I think is a particularly good way of explaining it.]
Re: What is the inverse of a vector?
#8While 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 for a long time. (https://en.wikipedia.org/wiki/Division_algebra)
When studying math at german universities, alebras are something you'll encounter in your 2nd year (latest; but might already show up in 1st year analysis albeit with a different focus). Implicitly, division algebras show up a lot when students learn about field extensions, galois theory and the algebraic closure of a field (usually 3rd semester). A more general treatment of division algebras is not a common subject, though.
Re: What is the inverse of a vector?
#9Ah, 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…
If you are working in ordinary Euclidean space with orthonormal bases, it doesn't do much for you. When you start doing calculus on embedded surfaces (the beginning of differential manifolds) it begins to be more helpful. Since Lie algebras are special types of differential manifolds, you can learn quite a bit by studying the geometry and this quickly leads into gauge theory and modern physics. The Geometry of Physics: An Introduction, by Theodore Frankel does a good job illustrating a lot of aspects of geometry even offering some geometric insight into some classical physics.
It is an absolute necessity for general relativity.
Re: What is the inverse of a vector?
#10Ah, 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…
> 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…
No, a vector is defined as an object that has certain properties, like addition and scalar multiplication. It's a very general, and abstract concept.
There are vector spaces of functions, with infinite dimension, but there are also vector spaces with a finite number of elements.
So only some vectors can even be written as 1xN matrices, if that is what you're referring to. But even if you write a vector that way, it doesn't mean it IS a matrix or that it automatically "has" multiplication.
In mathematics, an object only has an operation if it's part of the definition, and as such, vectors don't "have" multiplication.