What is (a) Space? From Zero to Geo 1.5
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- Опубліковано 16 чер 2024
- What is space? In this video, we learn about the many different things that we might call "space". We come up with both a geometric and an algebraic definition, and the discussion also leads us to the important concept of subspaces.
Sorry for how long this video took to make! I mentioned in a community post that I had personal things interfering with making this video, but even after that was out of the way, this video was really hard to make. I kept rewriting parts of it to be less confusing, and a few changes I made required drastic modifications to whole video (for example, I was originally going to define linear combinations here but then decided to wait until the next video, but I had used linear combinations in the rest of this video so I had to remove all references to them). Hopefully the video is worth the wait.
This video is a part of "From Zero to Geo", a series where we formulate geometric algebra, an incredibly powerful branch of mathematics, from the ground up. Full playlist here: • From Zero to Geo
Next video here: • Describing Many Vector...
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Sections:
00:00 Introduction
01:05 Describing regions with vectors
01:52 What is space?
03:24 Geometric definition of a space
04:40 Algebraic definition of a space
08:41 Basic consequences of the definition
09:32 Subspaces
10:13 Line exercises
11:54 Half-plane exercise
12:32 Scalar multiple exercise
14:21 Unit vector exercise
14:43 Zero vector exercise
15:35 Subspaces of all spaces
16:03 Classifying all spaces
16:43 Conclusion
That's a very nice intuitive explanation of where do the axioms come from and what they fundamentally mean.
Yea unfortunately some professors don't go over this stuff but that's understandable cuz they are required by school to go over all the curriculum in such short period of time
Finally, the wait is over
I thought the exact same thing
Strictly speaking, the vector space over a field K must have it that K obeys the field axioms, where you must also check if K obeys the group axioms. Then there is that linearity and distribution axioms of vector space, neutral element, inverse element, depending on the action
This video is implicitly assuming that we are talking about Euclidean vectors where all of the algebraic axioms are already satisfied. I'll cover more general linear spaces in a few videos.
a thing that has happened in the past few months since this august when the series started is im sudying math in university, and all the topics are something i've had to study now.
still love the uploads!
This is insane - I always "thought" of spaces in the way you explained, but I was never properly able to word it like that.
Also, the mixing of geometry and algebra is always so pleasant in your content! I hope you keep up the great work, as I believe your content might become the modern standard for teaching even linear algebra, clear, concise, visual and full of practicality built into it
It's so pleasing to see a new video in the series being released
Btw these 17 minutes passed just like 4
Super interesting series
Fantastic explanation. After many years of using linear algebra, this is the most straightforward (and accurate) explanation of this.
Only God can reward you. We are expecting others shortly. Thank you Prof
loving these videos! Have been reading about geometric algebra for a number of years and this is one of the nicest explanations I've seen!
This is so good; keep it up! I can't wait for it to finish so I can go back and binge it all again!
These videos are amazing, please don't stop making them!!!
i'm loving this series! very intuitive and pleasing to watch.
I'm very impressed by the quality of this explanation. I can't wait for the next video.
Wait. Is there really gonna be 5 entire chapters of this? Seems like its gonna stretch for a while :D
There's going to be 7 chapters!
@@sudgylacmoe It's okay, sudgy just described how this basis can stretch to an entire series and still be in the space. It'll all work out well. :-)
@@JohnSmith-ch9sm that was a really good one :3
7 chapters at this pace? Please hurry! I'm 77 years old.
Great video! I really like that you present the intuive ideias first and then, based on them, show what the axioms of vector spaces should be. Also, it's nice that the exercices you propose in the video are pretty simple (and perhaps trivial for those at university), so that people with few knowledge on it can uderstand. Btw, it feels like we are going to pass through a lot of linear algebra. In what chapter are we going to see new concepts of geometric algebra?
Chapter 1 is going to be pretty much pure linear algebra (but just vectors, no linear transformations yet). Chapter 2 will introduce multivectors, which are new for most people, but some might recognize them from exterior algebra. Chapters 3-5 are pure geometric algebra. Chapter 6 will cover linear transformations, which is a linear algebra topic, but with many of the ideas from geometric algebra incorporated into it (such as an incredibly simple definition of the determinant). Chapter 7 won't really be introducing any new concepts.
Nice Presentation sir..
15:46 the answer I came up with was that if you have two vectors v and u, the space containing all av's and the one containing all au's are both subspaces of that is a subspace of all linear combinations of av+bu
Awesome! These are the best introduction I have found! The future is geometrical andtopological!
Honestly the way I see it there was AdS space, fully expansive with a pressure to that expansion but in perfect balance. Then energy input, which created divergence which created convergence and then the entire surface of the universe was trying to be at a single point. But identity prevents this, so an infinitesimal surface of time was created so that infinity points can all share a moment together and then another new infinity points shares the surface of NOW, and this is how the universe is. The skin of infinitesimal time. Matter on one side of membrane and antimatter on the other. The surface of NOW. An inflow here is an outflow from there. Clockwise away here is counterclockwise towards there. This is why chirality. Why electron half spin? One evident orbit on this side and one internalized orbit on other side.
An inflow/divergence=negative charge (arbitrary convention notwithstanding) and outflow/convergence=positive charge. This is why the fundamental mass particle has positive charge. Gravity also is convergence but not of charge flow but of density. Energy density.
Awesome video
I'm especially looking forward to the video on direct sums
I really got it thank you so much
Great Video
I'm really liking the proposal of this series and I understand that at this point some conditions are being left implicit so as to not clutter the explanation, but, just so I know I'm following it, if we are to take vectors as (non-empty) equivalence classes of oriented segments of same lenght and orientation (so as to allow for freely moving around a vector), in order for the half-space to not count as a space we must add the condition that all vectors have a representative starting at the origin, correct?
While you can go the equivalence-class method of formalizing this, I personally prefer formalizing it as just "A vector is an oriented segment that starts at the origin" and then use the freely moving around property just for geometric intuition.
More specifically this video is talking about vector spaces aka linear spaces. I thought it was gonna talk about more general stuff like metric and topological spaces.
Very clear, concise and accurate video though. Nice work.
Could you do a similar video on "What is a structure?"
Is the empty set a space? Both restrictions are true an all of the vectors in the empty set (because there are no vectors). Also, if the empty set is a space, then it would be a subspace of every space, and the set containing the 0 vector wouldn’t be a subspace of the empty set.
Great question! I didn't mention this because I thought that most people wouldn't think of it and that it would be an unnecessary complication to the video. Because the empty set doesn't contain the zero vector, we wouldn't want it to be a space, although it actually would be a space according to the definition I gave in the video, as you pointed out. Another condition that is usually added is to require that the set is nonempty. Some people instead explicitly say that the set must contain the zero vector.
In more formal and rigorous Linear Algebra, the empty set is not a vectorspace
In more formal linear algebra, linear spaces are required to have an additive identity, so the empty set is not a linear space.
@@sudgylacmoe yeah sorry, I mean to say not. Or I meant to say that the element with only 0 is a trivial subspace. Thanks for the quick response
Thank you sudgylacmoe and ShadowZZZ
Thank you.
I was watching some of your videos on geometric algebra, and my linear algebra - I haven't used in about a decade but I feel like big chunks of it came back. One thing I was thinking while watching the video talking about the definition of a space was fractal dimensions / hausdorff dimension (I had watched a 3blue1brown videon on the topic not long ago). It made me wonder whether there was something analogous to vector spaces for partial 'dimensions' that show up with fractals. I briefly googled 'fractal dimension geometric algebra' and found something on 'fractal clifford spaces' and I am not entirely sure if it is what I was looking for. I am curious if you know of places to look for more info. Thanks!
I've never heard of fractal Clifford spaces so I can't really help you there. In general, there are many different definitions of dimension that are useful in different circumstances. The linear algebra definition of dimension only allows for dimension to be a cardinal, which discludes fractional dimension. I can't think of how to mix either linear or geometric algebra with Hausdorff dimension.
If you're not sure then I'm definitely not sure, lol. oh well! I'll play around a bit and probably not get anywhere but maybe learn stuff. Thank you!
Are you going to talk about ∞-dimensional spaces at some point?
Also, are you going to talk about dimension and co-dimension at some point?
For example, I found that subspaces with the same co-dimension have some things in common when it comes to intersection in the finite-dimensional world, but I don't know if that holds up on ∞-dimensional ones. As an example, the polynomials (with real value coefficients) are a space and the even and odd polynomials, respectively, form subspaces that have the same dimension as the parent space.
I will talk about dimension. This series is supposed to focus on geometry (it's geometric algebra after all), so I won't really talk about infinite dimensional spaces (I'll mention them just to say that they exist, but they aren't used in basic geometric algebra). There are some ideas in geometric algebra that relate to codimension, but I will not be using the term.
Finally!, yeeeah!, thanks!
Perfect
I think it's a bit of a stretch to call vector spaces just "spaces". The name "space" should belong to any topological space, at least. Many commenters below are confused from what I've read.
In spite of this didactic disagreement, I love your work! I love math and I love watching it on a big screen, with aesthetic visuals.
Yeah, I feel bad about this. I was thinking about adding a clarification about this but I forgot to do it. However, what I'm talking about in this video is not all vector spaces. In this video, I'm asking, "which Euclidean vectors are a vector space?", so saying that this video is about "What is a vector space" would be incorrect.
04:35 Can we have a space like a spherical shell that has a region but no origin within it? 9:30 mentioned about "closure" i.e. as a group, the zero (and identity) element should be part of the group. Bat that depend upon how you define the coordinate. you can have say a vector on a spherical shell to be a zero vector and the original and all operator operate on the shell of this sphere. Still wonder. BTW, the closure idea explained quite well between 8:30 and 9:30.
No, because every space must contain the zero vector.
If you want a spherical shell without the origin, you can define it as a continuous group. The elements of the group would be vectors of length 1 and the operation of the group is rotation.
Legend :)
Does this work for non-euclidean stuff too?
Like for example the half plane from 12:00, but (0,y)=(0,-y) and everything is connected on that axis like a portal or something. Wouldn't this also be a space? Because then the negative of any vector would be on it still for example -2 * (2,3) = (4,-6).
And what about the 1d geometry of a circle with a circumference of 1 the origin on it somewhere. Basically a line with a finite length and the two ends are connected together. Scaling would just wrap around on itself and adding would to for example 2 * (.5) = (0) and 3 * (0.5) = 0.5 and (0.5) + (0.1) = (0.6) and (0.3) - (0.4) = (0.9). So Wouldn't that be a space?
And if that later example is a valid space then wouldn't that mean that the part says that it needs to infinite in every direction is wrong? Instead it should be that the are no operations that can get you to a location not on the space or something.
Just want to know if my thoughts are right.
In doing what you're doing, you are redefining the operations we do on vectors. I haven't thought it through carefully, but I don't think these new operations you are presenting would actually form a space.
Here's an explicit example of why your first idea doesn't work: for most vectors, v - v != 0. For example, (1, 1) - (1, 1) = (1, 1) + (1, -1) = (2, 0) != (0, 0). While this doesn't seem to be an issue with the definition of a space in this video, it is an issue with the formal definition of a vector given in section 1.9.
Basically, doing this makes the objects you are working with not vectors.
@@sudgylacmoe Yeah I realized that when I watched that video
Hey, you didn't mention -1 dimensional spaces! (There's nothing in the rulebook that says {} can't be a vector space.)
you should really do a series on group theory first
u said a space is a region with infinite points/vectors . the space with just the 0 vector doesn't seem like a space to me since it contradicts that definition even tho it agrees with the other 2 algebraic conditions . thus i feel we should specify that in those algebraic definitions , v vector should be non null
0 dimensional spaces dont count then
3:31 "infinite continous region"... that askes for one more question , is a point continuous? i feel its not right because for smth to be continuous there should be a left and right hand limit and that should be equal to the value around which we need to check continouty , here we don't have that
Good point on the infinite part. I probably shouldn't have said that because formally, the zero vector by itself is in fact a space. As for continuity, I was using the term in the colloquial sense, not in the calculus sense.
Basically, the entire purpose of the "geometric intuition" section was to lead up to the rigorous definition of adding and scaling vectors staying in the space. That algebraic definition is the definition that is always used, and it supersedes any intuition we might think we have.
Would definitely buy a Book from you on GA, even if its like 50,- €
👍
Wouldn't the second condition imply the first? To me, saying that for all v in a space P, λv is in the space too is equivalent to saying that v + (λ-1)v is in the space too, which directly follows from the second condition...
Consider all points corresponding to integers on a number line. For any two vectors in this set, adding them stays in the set, but this set is definitely not a space.
@@sudgylacmoe Thanks. If K is a field and V an abelian group, we require that the multiplicative group of K acts on the set V, so the 2 conditions (ab)v = a(bv) i.e mixed associativity and 1v = v must be satisfied for all a, b in K and v in V.
Additional compatibilities between elements and operators from these 2 sets are:
1) Postulated where necessary
2) Defined where possible
3) Derived where sufficient
K is expanded to K, +, 0, -
V to V, +, 0, -
Having specified K and V, we need to express one way or the other, the mixed multiplication compatibilities with the each from the expanded list.
Once we obtain the full list of Necessary conditions (of course with redundancies) we can reduce to sufficient and independent list with fewer conditions.
The large list of necessary conditions helps to decide which is not a Vector space by violating just any one condition.
The smaller sufficient list helps to verify fewer conditions to be a Vector space.
"In God we live and move and have our being".
15:37 what?🤔I don't get it...
a(vect~v) is a space,
|a(vect~v)| = 0 ,length, is not a space,
but |a(vect~0)| = 0 , vect zero is a space, I don't get it, but what I got from this is...
a(vect~v) is a space, but the length that it describe is not a space because it is a magnitude?
a(vect~0) , vector zero, IS a space because in context, it is a vector (an origin)
~I'm not quite sure though... 😥
So in set theory ( I know a little) would be like...
(universal set for 1 dimension )U = { 0-vector, a*v~vector | 'a' is a scalar}
(universal set for 2 dimension )U = { 0-vector, (a*v~vector+b*w~vector) | 'a&b' are scalars} v&w~vector would be more sensible with the name x~hat&y~hat.
A space is a set that satisfies some property. Thats it.
My guess: a space should be closed under addition and scalar multiplication
Is the surface of a sphere a 2d space?
If you were asking if the surface of a sphere is a subspace of 3d space, the answer is no. Take any vector on the surface of the sphere and scale it by a scalar that isn't negative one. Then you'll get a vector not on the surface of the sphere, so scaling a vector on the sphere does not leave it on the sphere.
However, there are some other ways to try to force it to be space, but they get more complicated than what I've described in this video. I'll get to one of those ways in chapter 5.
@@sudgylacmoe No, I was talking about, for example, a line (1d space) that comes back on to itself because it is embedded on a möbius strip or the surface of sphere or of donut as a 2d space. Basically, where it is possible a.v = b.v but a not = b (where a and b are scalar and v is a vector). For the line the vectors are 1d and the vectors on the surface are 2d.
@@dominiquefortin5345 These are all topological spaces. But in this series, the author talk about *vector spaces*, and a circle, Möbius strip or surface of a sphere are not vector spaces.
please fix subtitles
The empty place between my ears.
Echo!
If you get a notification saying you got a dislike, it was an accident I swear!!
What is space ? Space is "the final frontier " everybody knows that. Silly question.