Відео

I would buy the book too. 🤔

I think the main upshot of this short is that when you use the word dimension, define your context. *Unit* quaternions are isomorphic to SU(2), which is three dimensional, and diffeomorphic to the 3-sphere, also three dimensional. It's unit quaternions that we use to represent 3D rotations. But general quaternions can be identified naturally with R^4 or C^2, and those most certainly are four dimensional.

while the video says "inside" and "outside", it can be understood as spam

Is this related to why the two dimensional rotation formula doesn’t work in 3D?

Shouldn't the u-hat on the right be inverted too? Or is that unnecessary for reflecting vectors?

Wow this is genius 🙇

Easily the most helpful algebra (mathematics) explanation video I have ever seen on UA-cam! I will definitely watch this channel whenever I have a math problem to solve.

great series! it's all about the basis vectors! woop woop!

That's crazy! Good to know

what

n space, n reflections, then x many kinds of transformation? orthogonal, affine, are there more? what makes a reflection orthogonal and a reflection non-orthogonal?

Автор чего-то немного путает. Если парабола - это просто функция от 1 переменной с 3 параметрами, то кватернион - объект вообще другой природы. Конкретнее, кватернион - это вектор в четырехмерном пространстве с базисом (1, i, j, k)

Do the properties of vectors and spans covered so for ensure that spaces are flat? I suspect so, but I'm not certain why. I think linear combinations of vectors would be enough to cover any space. (I'm not sure I can prove that, but it seems to work fine for covering a sphere, for example.) I suspect instead that we implicitly made space flat with our definition of length and by assuming that every point in space can be associated with a _unique_ vector.

From the standpoint of the stuff I've been learning recently the amazing consequence of this is that every rotation in SU(2) can be represented as the composition of at most 2n - 2 reflections. It makes understanding spinors and Spin(n) a lot easier.

This doesn't make sense to me since ((v dot u) v) can only result in a scaled vector v, the result can never be be a vector that is reflected across v. Also if u and v is perpendicular the result will always be zero. Or is it another product and not the dot product?

Simple! Basic hypothesis! On reflection . . .

The one time visualisations actually hinder understanding..

Remove the graphs and voice over and add in some women in bikinis dancing and you'd have a pretty good video.

Nice proof.

So problem, this isn't very helpful. If you don't know any of this then this man is speaking gibberish, and if you do, then this is just like ya.

Instructions unclear, my junk got stuck in a toaster

Indubitably

Well, I can check this off my bucket list.

Now I know what it feels like to others when I try to talk about computer science stuff with them.

very slick! probably the biggest missing detail is how we know that an n-dimensional subspace is preserved under both maps. it's not too difficult to see this, though; given our arbitrary vector v, the orthogonal complement of v must be preserved since v is preserved and the map respects orthogonality. (i guess you need to show that reflections are orthogonal, but that's a pretty standard result.) the second big detail is that we can reflect v back to where it started, but this can be shown by analysing the subspace spanned by v and Av and hence reducing to the R² case. the final detail is the base case, which is pretty obvious if we take it to be R¹.

This was awesome. You should do a long video explaining the details of the proof. I teach analytic geometry in my university and we cover a great deal about orthogonal transformations. I will include this in the next term’s syllabus

This helps you see in pictures. You are able to connect different parts of physics with ease.

39:28 If magnetic monopoles are included, and can define a thing analogous to J (I’ll call it, uh… D?), combining magnetic density and magnetic current, and then the right hand side becomes J + iD I believe

the bivectors would have to be constructed in the right order, yes? or some might cancel out the others?

16:59 Correction: The circular sector is not subtended by the angle. The angle is subtended by the arc boundary of the sector.

Ok this is actually beautiful

The vectors you are describing belong to real vector spaces (elements of R^n). Does it mean that complex vectors (elements of C^n) are not considered in your construction of geometric algebra?

Span implies that you have a plane shape, how do we go from plane -> point? 9:44

5:32 this is a special case I believe. Only true when |a| = |b| = |1|

probably a stupid question, because i dont know a lot about algebra, but what is 'vanilla' geometric algebra?

Hey when will you start posting videos of "from 0 to geo" again? I've been waiting for so long. I just wanted to ask.

This is equivalent to negative conjugation in SU(2) with Pauli matrices, right?

Nice work, but this isn't a good introduction to hestenes work. Everybody uses rotation matrices for lorentz transformations already. I think a really awesome video would be to take his STA to elucidate the Dirac equation as he does in his paper.

Is this actually the best way to do it?

This is so well done, man, congrats for your talent.

I feel as if im peeking into a cheese tactic for speedruning physics understanding any%

What about normalization tho

Ever heard of prime number space?

Wait so in 4D there is a Cube reflection? That's so insane I don't have the slightest idea how would that work!

...Is that the Dango from Clannad?

Following your reasoning, complex numbers, a + bi, are 1D. 😂 LOL

❤

A reflection is an analogue simulation.

oh wow, didn't expect to see the de morgan law here