What are...tensor products?
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- Опубликовано: 6 фев 2021
- Goal.
Explaining basic concepts of linear algebra in an intuitive way.
This time.
What are...tensor products? Or: How to multiply vector spaces and matrices.
Slides.
www.dtubbenhauer.com/youtube.html
#linearalgebra
#algebra
#mathematics
I'm currently taking Lin Alg 2 (math 540 at UW Madison) and we are using Paul Halmos's Finite Dimensional Vector space for this class. This is exactly how Halmos motivates tensor products but u not only cleared a lot of confusions I had while reading the textbook but also offered some really nice concrete examples!
Edit: My biggest frustration with tensor products is that there aren't many beginner level literature of this subject and it's very debilitating (in terms of motivation to learn) for ppl who just encountered them
Awesome, thanks for the feedback. I am glad that the video was helpful! Enjoy your linear algebra journey since it is actually enjoyable ;-)
A simple way to motivate tensor products is to consider the usual product of two univariate polynomials, each in a different variable (Professor Macauley's video on tensors mentioned that).
@@mingmiao364 That is another cool motivation, indeed! Thanks.
Thanks for this handson explanation, it helped me a ton!
I am glad that it was helpful! It also helped me a lot when I first came across this explanation.
Thanks for a very informative and easy to understand explanation. Much appreciated.
Thanks for the feedback. Tensor product are most often defined very abstractly, which is not for everyone, including myself. I really do hope that my rather explicit approach to tensor products in this video was helpful. Obviously the explicit way of defining tensor product is not due to me, but when I first learned it was the Eureka moment for me. Hope that worked for you as well!
This was so clear!
Glad that my explanation worked for you! Tensor products went all the way from very explicit to very abstract, so I hope the more down to earth explanation is useful for someone! When I heard this for the first time it worked for me, so I hope it works for you as well.
I guess i am 2 dim for this!
I don't think so.
I would just give it some time. For example, I never understood tensor products and was convinced I never would. So I stop trying. After a while I revisited them and here we are. (Not claiming that I understand what is going on...)
Reading or hearing different explanations also helps a lot - maybe the way I explain tensor products isn't quite right for you.
@@VisualMath I think the word tensor is too general - a tensor can be just a vector or a matrix or something more complex. Also I don't see where inner products and outer products fit in - are they just for vectors or are they also types of of tensor products? The terminology is confusing. It would help to see the tensor product of two 2x2 matrices.
@@MrSidTaylor I agree that the word itself is used in many different context, not always meaning the same thing. I am using "tensor product":
en.wikipedia.org/wiki/Tensor_product
A discussion of terminology can be found here:
math.stackexchange.com/questions/2624411/mathematicians-tensors-vs-physicists-tensors
I am naive: for me a "tensor" is nothing sophisticated but simply a vector in a tensor product. It is also a functor, so it takes matrices as input and returns matrices. For a 2x2 example, say A*B, replace any entry a_ij of the first matrix A with the block matrix a_ijB.
now let's add scalars and tensors :D
Sounds good - I am on board!
@@VisualMath thank you for all the videos you make! they help me a lot, especially the abstract algebra series rn. would it be possible for you to make a similar series on operator theory and operator algebras?
@@mastershooter64 Thanks for the feedback and the kind words.
I like the idea "What is...operator theory?", but nothing is planed right now. So I can't promise anything (life makes strange turns all the time - I am sure you can relate ;-)).
@@VisualMath I sure can relate, life is a rollercoaster!