Tensor products

I discuss tensor products.

[@edoardostefaninmustacchi2232]

I don't know what it is but these two videos made dual spaces and tensor spaces click for me for the first time after a long struggle. I am almost emotional, thank you.

[@kisonecat]

You're so welcome. Linear algebra is a source of frisson!

[@dewinmoonl]

thank you for this great lecture. It's hard to find short/clean intro to tensor product from purely a math view point rather than involving physics and mechanical engineering.

[@kisonecat]

Thanks! I'm glad you liked it. I'm thinking about producing more multilinear algebra materials...

[@phmfthacim]

Thank you, I've been looking all over the place for this stuff. Wikipedia, IRC, google, etc. This is clear and to the point.

[@orchidion7634]

Excellent! This truly conveys the core idea of the tensor product.

[@jpowerjalt]

Thank you so much for these videos! They're really helping me through my multilinear algebra course.

[@amitofu]

Thank you. I wasn't quite able to put the pieces together about tensor products from Wikipedia alone. This video gives me the foothold I need to continue. I hope you make more videos like this. I really like your concise, methodical style. Thanks again.

[@sourcreampotatochips]

This was awesome! It explained tensor products so much better than... pretty much any other source I could find out there.

[@kisonecat]

I'm glad you like them!

[@kisonecat]

Yes, as long as you are only talking about V circle-plus W and V times W. If you were to take a product or sum over an infinite indexing set (e.g., if I had a vector space for each natural numbers, and I circle-plused them all together), then these are very different.

[@StratosFair]

Great lecture ! Thank you

[@jonathanw2128]

I finally got it 😍 Thanks

[@dodeadson]

A very good set of slides. You've explained the motivation behind tensor products very well and I now finally understand what they are about. Just one thing to mention: you seem to switch to the next slide whilst you are still talking about the contents of the previous slide. It would be much easier for the viewer to follow if you were to only transition to the next slide/block of text when you have finished talking about the current slide/block of text

[@kisonecat]

Thanks! You are absolutely right that I need to fix the slide transitions.

[@nanotekman]

Thanks. That was helpful.

[@krishna058]

Thank you for the video. Can you suggest a link to show us an example of this. Also we took the "U" and 'V" as vectors (or tensors of rank one). I was looking forward to learn about rank 2 tensors. Any help on this will be greatly appreciated. Thank you once again.

[@saberking7869]

Great vid, thanks!

[@levigomes889]

👏👏👏

[@psydunk1]

hey jim, why dont you put it in the MV class!!!! it wud be very good!!! awesome Video btw!!!!!

[@alexj6935]

very good job. Thank you

[@kisonecat]

You're welcome!

[@sahmaddast3655]

For the last questions, are components of u tensor v linear maps from u to v?

[@viewer11004]

Great video!! can you please do a video on Kronecker product, thanks in advance :p

[@Patsoawsm]

Wow, I just finally got the idea behind tensor products. Thanks a lot! To the 'final puzzle', the only reason I could think of is that you could interpret Hom(U,V) as the ring of all dim(U) x dim(V) matrices, which naturally has dimension dim(U)*dim(V) but that only works for finite dimensional vector spaces. Is there a 'prettier' answer to that question?

[@ixem7985]

Vector spaces are free module and tensor product of direct sum is a direct sum of tensor product

[@retreatingtactic]

Then how do you relate the tensor product of U and V to the ring of all dim(U) x dim(V) matrices?

[@arturopresa]

Little Correction: Around the minute 5:16 you said any element of the tensor product of R^2 with itself is a sum of the four vectors mentioned, but rather than a sum, it should be a linear combination.

[@jacksong8748]

I noticed this. (I'm really just learning this stuff for the first time, so take it with a grain of salt, but) I was trusting every word he said up until this point and I found my self rewinding and listening over and over, until i realized that he misspoke.

[@DrPatZMusic]

Nice!

[@NoNTr1v1aL]

Thank you so much! I have a modules test tomorrow.

[@kisonecat]

Have fun on your test today!

[@gulxayosolijanova6018]

Thanks!

[@baohieu0610]

Can you please give me the PowerPoint file ? How you do that is so beautiful video? You can guide yourself not it? Thank you very much

[@yt-1161]

So what was the difference between bilinearity and linearity in those examples

[@markusluftner8418]

i think this brings it on point but a bit hard for newbees, try to introduce the idea of bilinearity and tensors first before you go into tensorprodukts and also please work on your audio qualety. But never the less your content is very good keep it up ;)

[@willful759]

I don't really understand a lot about category theory, but dim(U x V) = dim Hom (U,V) sounds like the yoneda lemma to me

[@bohlokoamile991]

🌻thank you

[@kisonecat]

You're welcome!

[@ayatnassar873]

Thank you

[@kisonecat]

You're welcome

[@ARBB1]

Nice

[@kisonecat]

Thanks! I'd like to do more tensor stuff at some point... maybe even this fall since I am scheduled to teach a differential manifolds course.

[@ARBB1]

@@kisonecat ah that's very good. You should keep at it definitely

[@ismaraabbasi4397]

Tell me five vectors and five scalars which are not tensor

[@michaelhiltz7846]

Is it fair to say that the tensor product is just each part of the vector u being multiplied by both all of the parts of the vector v? For example lets say the vector u = , and v is . Then u tensor v would equal = ?

[@kisonecat]

Great question -- this operation is sometimes called the "Kronecker product". I wouldn't say the tensor product is "just" this though, for the same reason I wouldn't say a linear transformation is "just" a matrix -- there's a choice of basis involved!

[@aix83]

Around 15 seconds in the video, why not be able to give an actual definition? Because it's a short video, or...?

[@parth3876]

Because tensor definition is complicated.

[@theskycuber4213]

the tensor product's definition is very abstract and is best understood when read, first one needs to understand how to formally motivate the definition by understanding the universal property, then define the tensor product using a "formal vector space", or a quotient on a bigger space

[@user-ov7lq7gh7i]

@@theskycuber4213 can you answer me about some tensor questions

[@JohnCurry1]

This is too abstract for me. I was hoping for a simpler explanation. For example, if you have two vectors V1 and V2, do they have a tensor product and what form would it take?

[@jasonbroadway8027]

I mostly enjoyed the video, but you lost me at 1:32. I did not follow the "subsume" idea.

[@naardebioscoop]

horrible clipping on audio, dude, turn down your input gain

[@naardebioscoop]

(but decent lecture)

[@ChaseWadeGoodknight]

horrible communication skills, dude, tone it down a bit

[@ChaseWadeGoodknight]

​@@naardebioscoop(but decent advice)

[@michaellewis7861]

Poor explanation but nonetheless sufficient for my purposes.

[@kisonecat]

It is an old video which I would like to update someday soon. Let me know what you'd like to see in a video about the tensor product.

[@abublahinocuckbloho4539]

explaining the abstract properties of a tensor product doesnt actually explain what a tensor product is. this video is a kin to give an example of a concept rather than defining clearly what the concept is. this video could easily start with a basic understanding of what tensor products are with the tensor product between basis vectors e1 and e2 by actually going through the calculations.

[@MuffinsAPlenty]

"explaining the abstract properties of a tensor product doesnt actually explain what a tensor product is." It actually does though, since those properties pin down the tensor product up to isomorphism, and you can only define the tensor product up to isomorphism anyway.