But what is a Vector Space?

Vectors are fundamental tools in mathematics and sciences. Yet different fields like mathematics, physics, and engineering seem to define vectors differently. It is mathematics, unsurprisingly, that formally defines a vector. We will go through this process of formalization, using the foundational tools of abstract algebra to define and construct a vector space.
Additional Resources:
Wikipedia article on fields: en.wikipedia.org/wiki/Field_(...)
Wikipedia article on vector spaces: en.wikipedia.org/wiki/Vector_...
Linear Algebra, 4th Edition by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence
Music:
c418.bandcamp.com/album/dief
Keighley by C418
Blank Cubicle by C418
c418.bandcamp.com/album/life-...
Life changing moments seem minor in pictures by C418
Droopy like your face by C418
patriciataxxon.bandcamp.com/a...
Minute by Patricia Taxxon
patriciataxxon.bandcamp.com/a...
Careful Beaver! by Patricia Taxxon
Animations were made by Manim, an open-source python-based animation program by 3Blue1Brown.
github.com/3b1b/manim
This video was submitted to 3Blue1Brown's SoME3 (Summer of Math Exposition 3).
some.3b1b.co/

[@aidsan6950]

This is a great video! Please keep these up. This is what math education should be, I think.

[@angeldude101]

Something I love about vector spaces is that... who said scalars aren't themselves vectors?🙃 If you want to be technical, it's possible to argue that ℝ is itself a vectorspace over the field ℝ, and, thinking about abstraction, the same should probably be true for any field.

[@adityaruplaha]

It is indeed true.

[@Happy_Abe]

And to abstract this a bit more, any field K is a vector space over any subfield F in K. The size of its basis over a subfield is what is known as the degree of the field extension from F to K. This is a huge concept in field theory and Galois theory.

[@dailymemigzugxoyditsi3273]

Brilliant 🎉 video. I would like to make a suggestion, that being a video on tensor just like this having axiomatic definition.
The reason I request is most of us students are clueless about what tensors is in mathematical stand point.

[@quantumgaming9180]

Unbelivably great videos as always. Although I like this video a lot, my favorite one is still the "Cardinality of the Continuum" video 😊

[@aincrad6383]

brilliant video

[@davidwright8432]

Many thanks! I'm giving myself a refresher, from the beginning, of stuff I 'learned' as an undergrad - a while back! Now it's all beginning to make sense - not just a way of doing problems by routine.

[@defango]

Awesome video!!!

[@martinsanchez-hw4fi]

Waiting for the next video. Bravo

[@JJ-tp2lc]

This isn’t related to any of the math, but I’m glad to see C418’s non Minecraft music being used

[@modolief]

Superb!

[@54johndavis]

Great info but the muzak is distraction.