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What is the inverse of a vector?

317 点作者 mkeeter超过 3 年前

32 条评论

Garlef超过 3 年前
While the article is written very nicely, It seems that this is written out of a perspective of some missing knowledge.<p>The basic object that the author seems to be interested in is that of an &quot;algebra over a field&quot; (<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Algebra_over_a_field" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Algebra_over_a_field</a>).<p>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. (<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Division_algebra" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Division_algebra</a>)<p>When studying math at german universities, alebras are something you&#x27;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.
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spekcular超过 3 年前
The article begins:<p>&gt;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.<p>and concludes:<p>&gt; 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&#x27;s notation for Calculus is no longer the dominant one, or that Maxwell&#x27;s actual equations for Electromagnetism have been replaced by Heaviside&#x27;s, textbooks will change because a better system has come along.<p>These claims are wrong.<p>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&#x27;m oversimplifying a bit, but a full discussion is too long for a comment here.]<p>Anyway, geometric algebra is not some esoteric secret. People know about it and have decided not to teach it, because the stuff that&#x27;s already taught is better.<p>[I picked up this specific phrasing from another user here, knzhou, which I think is a particularly good way of explaining it.]
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Grustaf超过 3 年前
Ah, another geometric algebra evangelist? I can&#x27;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.<p>As to vectors, obviously they have inverses, additive inverses. Since vectors don&#x27;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 &quot;the inverse of a vector&quot; anymore.
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ur-whale超过 3 年前
Geometric algebra (Clifford Algebra) unfortunately came late historically.<p>It&#x27;s a shame, because the whole theory is a very useful (eg for engineering &#x2F; applied math) superset of linear algebra.<p>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:<p><pre><code> differential forms tensor calculus linear algebra etc </code></pre> From zero to geo is a very good video introduction to the topic:<p><a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=2hBWCCAiCzQ&amp;list=PLVuwZXwFua-0Ks3rRS4tIkswgUmDLqqRy&amp;index=2" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=2hBWCCAiCzQ&amp;list=PLVuwZXwFua...</a>
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jefftk超过 3 年前
<i>&gt; The similarities are so striking that we might think of them as &quot;pseudovpseudovectors&quot;. But I won&#x27;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!</i><p><i>&gt; 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><p>I&#x27;m confused here and don&#x27;t understand why they keep saying a bivector has five degrees of freedom. If you can uniquely identify one with three scalar coefficients, doesn&#x27;t it only have three degrees of freedom?
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BlackFly超过 3 年前
For those who are interested, this sort of algebra would be known as the [Grassman algebra or the exterior algebra](<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Exterior_algebra" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Exterior_algebra</a>). 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.
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deepsun超过 3 年前
English&#x2F;American style of explanation fascinates me.<p>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!<p>Why, oh why do you need to waste yours and reader&#x27;s time to introduce basic definitions, if any reader of the article definitely knows that? If they haven&#x27;t, they wouldn&#x27;t be able to read the first paragraph at all.<p>PS: Russian style of explanation is more like: &quot;Here&#x27;s the essence of my idea, maybe with some leading pre-definitions, but definitely without basics. If you are here, you probably is as curious as I am to already know&#x2F;heard of all the basics.&quot; In total, there&#x27;s more material, because it&#x27;s easier to write and read it, as author didn&#x27;t need to explain 101s to PhDs.
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OscarCunningham超过 3 年前
People might be interested in a similar post I wrote about dividing by a vector (<a href="https:&#x2F;&#x2F;oscarcunningham.com&#x2F;4&#x2F;dividing-by-a-vector&#x2F;" rel="nofollow">https:&#x2F;&#x2F;oscarcunningham.com&#x2F;4&#x2F;dividing-by-a-vector&#x2F;</a>) although I came to a different answer as I was considering arbitrary vector spaces rather than just 3D space.
fouronnes3超过 3 年前
As a programmer it seems to me that the number one problem of math notation is that it&#x27;s weakly typed. There&#x27;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.
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debbiedowner超过 3 年前
Doesn&#x27;t seem self consistent. He defines the ab multiplication as dot product plus &quot;extrusion&quot;&#x2F;bivector (which seems simpler to call convex combinations of 0,a,b,a+b). Then he says aa is a scalar, presumably because the &quot;extrusion&quot; is 0, but you can&#x27;t have this identity be 0. Just because it&#x27;s degenerate does not mean it&#x27;s 0. And the &quot;extrusion&quot; while not a plane is a line in his definition.
sgt101超过 3 年前
I 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 +&#x2F;- for things like word2vec, but what would multiplication and inverse mean in this context?
necovek超过 3 年前
I love the way this was presented (others have pointed out flaws with the article already).<p>I&#x27;d appreciate a post from Matt Ferraro on how this is built. Bonus points for including nice syntax-highlighted code &quot;widget&quot; for a cross between maths&#x2F;programming.
codeflo超过 3 年前
The writing is cute and the animations are nice, but none of it makes any sense. I stopped reading at<p>&gt; 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.<p>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.
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da39a3ee超过 3 年前
&gt; A scalar is a point on a number line.<p>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.
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fallingfrog超过 3 年前
It occurs to me that some of these axioms depend on how many dimensions the space has- in 4 dimensions, a vector would have 4 components, a bivector would have 6, a trivector would have 4, and a quadvector would have 1. And so on, in accordance with Pascal’s triangle.
foxhop超过 3 年前
Here is something similar I wrote a long while back as notes for my future selfs: <a href="https:&#x2F;&#x2F;www.foxhop.net&#x2F;vector-math-for-video-games" rel="nofollow">https:&#x2F;&#x2F;www.foxhop.net&#x2F;vector-math-for-video-games</a>
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portpecos超过 3 年前
I&#x27;m taking a first semester physics course right now, and we&#x27;re learning about Torque and Angular Momentum. I just finished a calculus course last semester.<p>Can someone tell me how I would use τ=r∧F on a physics problem for Torque?
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amatic超过 3 年前
I would say the title should be &quot;reciprocal&quot; of a vector. Right? The inverse of a function (f^-1) has an unfortunate notation equality with the reciprocal (x^-1), or multiplicatory inverse. Or am I wrong?
SantiagoQ超过 3 年前
Semantically, the inverse of a vector is something that has no magnitude nor direction.<p>I wonder what would that look like (mathematically), and what surfaces or fields would it create?
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gpderetta超过 3 年前
Great article. I had a similar kind of revelation when I learned about generalized linear models after failing to understand all the various statistical tests.
felipeqq2超过 3 年前
A rotcev
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haunter超过 3 年前
Wow the interactive 3D illustrations are awesome. Works perfectly on touchscreen, feels very natural.
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billfruit超过 3 年前
Is there a book that give comprehensive treatment of euclidean geometry, but using vectors?
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anotheraccount9超过 3 年前
&quot;It turns out that inverting a vector on its own isn&#x27;t well defined.&quot;
cannabis_sam超过 3 年前
I would have loved to implement this in Haskell as an exercise in uni!
pmarreck超过 3 年前
Would this potentially enable generalized matrix inversion?
foobarbazbarfoo超过 3 年前
love your writing, can anyone recommend blogs like these
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fractal618超过 3 年前
The inverse of a any positive or negative vector with an amplitude greater than zero points directly into your soul relative to it&#x27;s original amplitude and the how many regrets you have.
unixhero超过 3 年前
Somehow I got that first math explanation.
bsedlm超过 3 年前
where has this been all my life!?<p>on the plus side I found it now...
DreamScatter超过 3 年前
This article is really about geometric algebra, check out my geometric algebra software <a href="https:&#x2F;&#x2F;github.com&#x2F;chakravala&#x2F;Grassmann.jl" rel="nofollow">https:&#x2F;&#x2F;github.com&#x2F;chakravala&#x2F;Grassmann.jl</a>
jeffwass超过 3 年前
I stopped reading at this paragraph near the top :<p>“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.”
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