What is...an affine space?

Рет қаралды 37,332

Күн бұрын

Пікірлер: 74

@thinkmachine_ 3 жыл бұрын

I was looking for an intuitive explanation on affine spaces and found this video, I thank you for all that you put into making these videos.

@VisualMath 3 жыл бұрын

You are welcome!

@curiouscraw3927 2 жыл бұрын

Great video, helped me to finally get a sound understanding of what my professor means when he describes affine geometry as geometry where they lost the origin. Thanks!

@VisualMath 2 жыл бұрын

Glad that the video was helpful! And I hope you haven't lost your origin, but only geometry did ;-) Just kidding, "the lost origin" is a good way to memorize what affine geometry is all about, and it worked great for me when I was in your situation. Having that idea explained in different words (or any idea actually, which is where KZbin can be really helpful) is the way to go, I guess. Glad that you feel the same, and my way of explaining the lost origin was useful.

@ivanfilippenko 15 күн бұрын

Another typo: at 2:12, in order to match the graph, equation (2) should evidently be x + y = 1/2

@VisualMath 15 күн бұрын

Thanks for catching; I have put a warning in the description.

@Agus-of6rh 2 жыл бұрын

Thank you, I needed some visual explanation to feel the subject "more alive" (showing "movement"), and your video did that for me.

@VisualMath 2 жыл бұрын

Glad it was helpful! The explanation worked for me, and it is good to know that you feel the same.

@Karim-nq1be Ай бұрын

a + 0 = a (small typo at 9:45). Thank you, always a pleasure to come back to your channel :-)

@VisualMath Ай бұрын

Ups, yes you are right. Thanks for catching ☺ I have put a warning in the description.

@vivaangupta4078 Жыл бұрын

The first matrix (on the left) at 10:49, I believe, doesn't correspond to the picture associated with it. The "linear part" of the matrix given is a -π/2 rotation (counter-clockwise) and it appears to me that the picture associated with it in this video is a π rotation (counter-clockwise).

@VisualMath Жыл бұрын

Ah, thanks for spotting that typo. You are right, and the correct green matrix would be {{-1, 0}, {0, -1}}. Thanks again, I have put a warning in the description.

@anandpaladhi8944 3 жыл бұрын

thank you...this helped me to understand the concept in much better way...though, could not follow "Matrices for affine maps" at 10.48 but probably need to learn little more and come back here again to revise...

@VisualMath 3 жыл бұрын

“but probably need to learn little more and come back here again to revise…“ I am almost 100% sure that you shouldn’t blame yourself: Most of the time it is the lecturers fault if the audience can not follow, so I am the one to blame! Let me try to motivate "Matrices for affine maps" at 10.48 differently. - In an affine space you need “one more coordinate than in the linear case” since you need to keep track of where vectors are anchored (thinking of vectors as arrows). In the linear aka non-affine case vectors are always anchored at the origin, so you do not need this coordinate. - Thus, an affine matrix for 2-space should be a 3x3 matrix, as on the slide. For n-space it should have (n+1)x(n+1) entries. - If you look at the involved notions in detail you will see that one actually does not need “all of” the 3x3 matrices, but the part marked on the slide suffices. - Then its a matter of convention what is the linear part and what is the affine part. On my slide the linear part is the 2x2 block to the northwest: whatever matrix you put there describes a usual linear transformation which fixes the origin. - With contrast, the affine part is the 2x1 block to the northeast and describes where the origin moves when applying the affine matrix. I hope that helps!

@almspb 3 жыл бұрын

Well, proper and highly quality explanation. Thank you very much.

@VisualMath 3 жыл бұрын

Thank you for the feedback, you are welcome! But I had an easy life, affine spaces are great! I hope that I was able to communicate that affine spaces are really cute (and also useful, although I haven't explained that I guess ;-)).

@lowerbound4803 Жыл бұрын

Great Explanation as always 😻 Thank you!🙏🙏

@VisualMath Жыл бұрын

And you are very welcome, as always 😘

@MathCuriousity Жыл бұрын

Hey question if you have a chance: why are we taught vector spaces and then affine spaces? Is there a way to define an affine space without talking about a vector space? I feel like I need a different way of thinking about affine spaces and I don’t like the idea of using vector spaces minus some vector property to get affine. Help me please understand a definition of affine space without vector space!

@VisualMath Жыл бұрын

Ah, a tricky one! The problem is scalar multiplication which should give you some element 0 in your space. But that is then something you do not want for affine spaces. Some ways around are described here: ncatlab.org/nlab/show/torsor math.stackexchange.com/questions/3664038 But its rather nasty. I hope that helps!

@MathCuriousity Жыл бұрын

@@VisualMath thank you I will check it out! The main thing which confuses me is ….we talk about an affine space as a vector space without an origin and to me that makes sense. It’s the formal definition you gave later on in the video where I figured we would no longer need this intuitive idea and would define an affine space as it’s own thing. Weren’t affine spaces “invented” before vector spaces anyway?

@VisualMath Жыл бұрын

With vector spaces and affine spaces its a bit as with "Chicken-and-egg"; not sure whether one can really say which one came first. My guess is that the bias that you see is that mathematics likes units, so vector spaces have a preference among mathematicians.

@MathCuriousity Жыл бұрын

One other question: could you explain what some professors mean when they talk of Euclidian affine space and euclidian vector space as opposed to just affine spade and vector space? Thanks so much!

@VisualMath Жыл бұрын

I think what they means by Euclidean is "over the real numbers with the usual scalar product". One usually refers to Euclidean space as something where one can do Euclidean geometry, which amounts to having real numbers and the usual notion of distance and angles. I hope that helps!

@MathCuriousity Жыл бұрын

@@VisualMath hey so I found another source saying basically if we say “vector space” we cannot assume that this space has length/distance/angles but if we say Euclidean vector space we can because it means “vector space with inner product”. The same for affine space vs affine space with inner product. Is that correct?! If so - Would an “affine space with inner product” be the same as a space that has vectors in an affine space (like intro physics diagrams when solving problems)? Thanks so much for taking the time to help me!

@VisualMath Жыл бұрын

Almost, Euclidean spaces are usually over the real numbers. Otherwise, yes.

@MathCuriousity Жыл бұрын

@@VisualMath ok cool so to wrap this up - for all intents and purposes - “Euclidean” added onto it means it has an inner product space. Right?

@VisualMath Жыл бұрын

@@MathCuriousity Yep, Euclidean = real inner product.

@xoppa09 Жыл бұрын

how do we go from an 'affine space' in geometry where there is no origin, to an 'affine transformation'. awesome video

@VisualMath Жыл бұрын

An affine transformation is a bijection endomorphism of an affine space that is an affine map, so linear in the appropriate sense. Examples include rotation, translation (this is not linear since the origin moves!), shearing... In other words, affine transformations are the analogs of invertible matrices in the affine world. I hope that clarifies the terminology.

@khoanguyen5321 Жыл бұрын

Is affine space a larger set than vector space? or it is like a 'dual' of a vector space?

@VisualMath Жыл бұрын

Neither of the two: - Affine space = vector space without origin, so "set-wise" they are of the same size. - No, that comparison doesn't quite work as the dual has an origin. Again, the slogan is "Affine space = vector space without origin". Otherwise there is no difference between the two. I hope that helps.

@khoanguyen5321 Жыл бұрын

So, there is no intersection between vector space and affine space, right? What I means is there is not exist a space that both be a vector space and affine space at the same time, right?

@khoanguyen5321 Жыл бұрын

The way that I think is the more structure, the smaller the set. Metric space is smaller than topological space because it is the topological space with additional structure which is notion of distance. If vector space is affine space with additional structure which is the origin, then vector space is smaller than affine space ??

@VisualMath Жыл бұрын

Oh, you want to compare the classes of vector spaces and affine spaces. I misunderstood. Ok, formally no affine space is a vector space and no vector space is an affine space. So the classes are distinct - two different universes. However, every affine space can (non-canonical) be made into a vector space and vice versa. Translation is the keyword. So the classes have the "same size" in a certain sense. But talking about sizes of non-set classes is a bit awkward, so maybe let us not do that. But I think it is fair to say that the two concepts are essentially the same. So not as metric and topological spaces that are honestly different.

@khoanguyen5321 Жыл бұрын

@@VisualMath Yes, that is what I mean when I said 'dual' above, they are the same "size" and can map to each other by adding or removing the origin. Sorry, for my bad interpretation, I don't know much about mathematical terminology to describe it properly. So, in terms of class size, topological space contains vector space, affine space, and metric space. There is no intersection b/w vector space and affine space. However, there is an intersection b/w vector space and metric space, and the intersection region contains Euclid vector space. Also, there is an intersection b/w affine space and metric space, and the intersection region contains affine Euclid space (aka Euclid space). We can also map from affine Euclid space to Euclid vector space by adding an origin. Is it correct?

@Victual88 5 ай бұрын

I watched it till the end; it's too good not to subscribe!

@VisualMath 5 ай бұрын

Haha, thanks. I am glad that you liked the video ☺

@jeanphilippepuyravaud439 3 жыл бұрын

Very useful. I saw the comment by AK. For me, your accent is fine, but you tend to lower your tone and volume at the end of the phrases. So it looks as if the important point is xxx.

@VisualMath 3 жыл бұрын

Great, thanks for the feedback. That is hard to change, of course, but I will see what I can do.

@mangaart3366 3 жыл бұрын

Thank you for this video it helped me a lot!

@VisualMath 3 жыл бұрын

Thank you for the feedback. I always hope that these videos will be of some use.

@minhnam3827 7 ай бұрын

Great video!!!! Thank you so muchhh

@VisualMath 7 ай бұрын

Thanks for the feedback, I hope you will enjoy linear algebra ☺

@jonathanlister5644 Жыл бұрын

Well, Thanks for your response. I have to say that I did find your presentation to be excellent, especially your graphical work. I suppose time is a factor in these presentations and you have to be selective in what you present as an introduction to Affine spaces! I did not mean my comment as a criticism. Yours Jonathan.

@VisualMath Жыл бұрын

No worries: I took it as positive criticisms, which is always welcome! We all should try to improve and your comments was very appreciated☺

@Labroidas 2 жыл бұрын

This is a great video, but if I may give a small critique: it's sometimes hard to understand what you're saying, mostly i think because the microphone you're using is pretty low-quality, the room you're in has a strong echo (the sound is reverberating off the walls), and you're speaking fast and are sometimes not enunciating enough. Nevertheless you're amazing at explaining, it helped me a lot, thank you so much!!

@VisualMath 2 жыл бұрын

Thanks for the feedback! The issue with the mic is solved in newer videos. I got a new mic. The old one was actually not bad. But it was not working properly and it took me a while to realize that - sorry for that! I hope the quality was still good enough so that the main points became clear.

@Labroidas 2 жыл бұрын

@@VisualMath Definitly! Sorry for bringing up a problem that is already solved, and thank you for your efforts!

@VisualMath 2 жыл бұрын

@@Labroidas No worries: I am not expecting you to watch all of my videos. Its questionable whether I did that myself ;-) So any feedback is welcome!

@jonathanlister5644 Жыл бұрын

Very interesting, however I rather think you have over-stated this no origin stuff, the point is there is no preferred origin. Also why not mention that contrary to the Euclidean geometry battered into our skulls at school, the distance between two points or the angle between two lines need not be significant. We should refrain from introducing either magnitude or direction as they are invariant concepts for us in this space.

@VisualMath Жыл бұрын

Thank you so much for the feedback. The points you make are important, thanks for posting! I decided to go with the explanation that has worked best for me, and this is what came out 😅