Dan Drake šŸ¦†Sep 3, 2023

OMG, where was this article when I was first learning about tensor products?

https://www.math3ma.com/blog/the-tensor-product-demystified

In 10 minutes I learned more about tensor products than I did in grad school. In particular, I see more of the connection between the way math uses tensor products and how physicist types think of them, as higher-dimensional matrices.

This kind of thing really riles me up. It's the kind of thing that makes me want to go back to teaching, because it hits the exact personality/psychological trait that makes me like teaching: I learn about some cool math thing and I want to charge into a classroom and say "GUYS! I just figured this out! It's so cool and interesting! OMG tensor products are the best, let me show you why!"

It bothers me that I was taught about tensor products in such a bad way. I want students to get something better than I did.

(OTOH, maybe I'm just not very smart, or was a bad student. That's a real possibility. But how great would it be for even dim bulbs and lazy students to get *something* out of learning about a topic?)

#math #linearalgebra #mtbos #teaching

The Tensor Product, Demystified

Show threadj_bertolotti
@ddrake Took me (many) years after graduating before a random comment by @johncarlosbaez made me finally understand what a tensor actually is. All the explanations I got as a student were so needlessly convoluted that nothing sticked with me.
Sep 3, 2023 at 12:31pmWeb
Show threadJohn Carlos BaezSep 3, 2023

@j_bertolotti - thanks, Jacob! I agree that tensors are often explained badly. And I agree with @ddrake that Tai-Danae Bradley's blog explanation is delightfully clear.

I learned tensors because I wanted to understand general relativity and this was a bridge I *had* to cross. So I spent a lot of time studying them. Nowadays Google's open-source machine learning software "TensorFlow" may push other people into understanding tensors. But it would be nice if people got a tiny taste of them right around when learning matrices.

Like: "Hey! It's cool how you can multiply a vector by a 2-dimensional array of numbers and get a vector. But it doesn't stop there! In a viscous fluid the stress is a 2d array of numbers and so is the gradient of velocity, and you have to multiply the gradient of the velocity by a 4-dimensional array of numbers to get the stress!"

Well, that's probably too intense, but just some hint that after Tįµ¢ā±¼ thingies we'll want Tįµ¢ā±¼ā‚– thingies and Tįµ¢ā±¼ā‚–ā‚— thingies and so on...

https://en.wikipedia.org/wiki/Viscosity#General_definition

Viscosity - Wikipedia

Show threadNikita LisitsaSep 3, 2023
@johncarlosbaez @j_bertolotti @ddrake I always understood them from the algebraic perspective, as the most general way to multiply vectors. This explains everything else about tensors so nicely!
Show threadJohn Carlos BaezSep 3, 2023

@lisyarus @j_bertolotti @ddrake - I probably started by understanding tensors from the physics perspective, where a rank-n tensor is a gadget that describes some numerical quantity that depends linearly on n different vectors. I might have seen them first in the Feynman Lectures.

Then we can refine this by distinguishing vectors from covectors. Then I learned about tensor products of vector spaces, and it became clear that the physicists' tensors are elements of V āŠ— ... āŠ— V āŠ— V* āŠ— ... āŠ— V*. This is what you learn in a math class, but also if you read a decent book on general relativity.

Show threadAaronSep 4, 2023
@johncarlosbaez @j_bertolotti @ddrake I came to an understanding through tensorflow. I wish I had had more of a mathematical exposure to them first, though.
Show threadDan Drake šŸ¦†Sep 4, 2023

@j_bertolotti I had the same thing happen with real analysis. I took it in undergrad, in grad school, and was always..."meh". But years later, after teaching basic stuff, and encountering little things about real analysis here and there, I came around to the "OMG, guys, this is amazing!" attitude.

Just like with the tensor product stuff that I started with here, I want to go back and teach real analysis, so that I can burst into a classroom and say, "guys! This amazing! The real numbers, and functions thereof, are SUPER WEIRD and amazing!"