What is the free vector space??

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Books I like:

Abstract Algebra:

Differential Forms:

Number Theory:

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[goodplacetostop]

0:25 Maybe it’s time to create a new channel? You can keep this channel for small exercises/contest problems and shorts and use the second one for more advanced topics, Q&A series, anything else? It’s an idea I just had

[mythicmansam]

Definitely preferr these types of videos. Not that the others aren't well done/entertaining/informative but these are just so cool!

[gucker]

I rarely watch your "contest problems" videos but I always watch videos like this one, where your explain various topics on algebra, number theory, etc.
Your explanations are great! Thank you!

[zuzaaa]

This and the yesterday video about tensor product were great videos. I love all your videos but these videos are especially entertaining because there were some new, abstract concepts that you presented in an easy to understand way. That would be great if you continued making this kind of algebraic videos. I hope that you could make them in such a way that to understand one video the viewer doesn't need to watch every single video in the series beforehand

[infphreak]

Instead of "formal linear combinations", I think it's simpler to define the free vector space over K on S as the set of all finitely supported functions from S to K.

[AntoshaPushkin]

I think it's much less confusing to look at the vectors of V(S) as functions v: S → R such that there is finite number of s in S for which v(s) ≠ 0, and adding two vectors v and w is just adding functions, multiplying by a constant is just multiplying a function by a constant

[brooksbryant]

More videos like this would be awesome, I thought the last tensor product one was really interesting

[nachiketjhala]

This is the most lifesaving video for me... every other content on the internet is so confusing and do not talk about the free vector space over a vector space... thanks for the vedio

[M.athematech]

The notation: sum for i= 1 to n, does not work when the empty sum is being considered. What works in general is: the sum for i < n. In the case where n = 0, there is no i < n so that case indicates an empty sum.

[KybursScrimmages]

Instead of talking about “formal sums”, you could say that the free vector space over R generated by a set S, V(S), is the set of all functions f:S—>R such that f(x)≠0 only for a finite number of x∈S.

To connect these two definitions you only have to identify every function f with the “formal sum”
Σf(x)x (x∈S)

[thegrandyata]

Please do more videos like these. I'm a math grad student so these higher level videos are actually really useful to me and I'm sure for a lot of others as well!

[l.p.]

These vector space videos have been great, I would really enjoy more videos about these. Not just vector spaces but maybe even fields and sub spaces too!

[romajimamulo]

11:11 maybe you could use like, putting the vector numbers in a box or do x subscript any real number?

[rafaelgcpp]

I really like algebra, and those "higher level" abstractions are really interesting.

On a different topic: do you have any videos on Galois fields?

[samallen]

I definitely prefer these kinds of videos as there is already a lot of competition style math videos out there and not really any videos like this.

[noahlibra]

Honestly I only watch your channel for these advanced topics. Keep them coming!

[madlarch]

I found this video *really* helpful after the tensor video. I was familiar with the free functor in category theory, but seeing this 'from the ground up' approach has given me a much better understanding of how that really works. Many thanks !!!

[ChefSalad]

I quite like these videos on abstract algebra, although I'd also like to see some videos on applications of abstract algebra or at least ones with good examples. This video certainly clarified the previous video, which is very welcome, although I thought the first one was pretty understandable on its own. And I'm very interested on the connection between these tensor products and the tensors used in general relativity. In fact, I think a decent explanation of the mathematics of those tensors and their operations would make a very good video series. I've read through several books on tensors and GR, and I never quite seem to fully understand what's going on. Recently I saw an explanation by another YouTuber that made everything make much more sense, but I still feel a bit in the dark on it. They explained tensors as basically objects that behave like vectors or square matrices in that they fundamentally stay the same under changes of basis, even though their elements are changed, but that tensors incorporate additional bits of information, like rank-3, dimension-3 tensors being like a triplet of triplets of vectors, in the same way that a dimension-3 square matrix is a triplet of vectors. Unfortunately, they never even mentioned co-variant and contra-variant tensors, and that's still a bit mysterious to me. I know the difference is in how they transform under a change of basis, but exactly what's going on is beyond me. I'd love it if you did a series explaining that stuff. The actual GR stuff is unnecessary, since the math is so interesting on its own.

[estuardodiaz]

Yes! more videos like this!! Awesome channel btw

[leotimm]

I just love this videos, thank you so much !!!