What is (a) Space? From Zero to Geo 1.5

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.

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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

[Pavgran]

That's a very nice intuitive explanation of where do the axioms come from and what they fundamentally mean.

[ShadowZZZ]

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

[pietrocelano]

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!

[kodirovsshik]

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

[daydreams]

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

[JohnSmith-chsm]

Fantastic explanation. After many years of using linear algebra, this is the most straightforward (and accurate) explanation of this.

[alamagordoingordo]

I'm very impressed by the quality of this explanation. I can't wait for the next video.

[_haru.o]

i'm loving this series! very intuitive and pleasing to watch.

[merseyviking]

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!

[Boringpenguin]

These videos are amazing, please don't stop making them!!!

[mateusmarchesi]

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?

[nicolaskrause]

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!

[A.Shafei]

I'm especially looking forward to the video on direct sums

[KaliFissure]

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.

[rasaqalao]

Only God can reward you. We are expecting others shortly. Thank you Prof

[smolboi]

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.

[GNeves]

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?

[keirablack]

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!

[dorol]

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

[mzg]

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.