Introduction to tensors in linear algebra

The study of linear algebra naturally leads to the study of multi-linear algebra. For example, inner products give examples of bilinear forms and are essential in the study of differential geometry and hence physics. The study of multi-linear algebra is best approached by introducing a gadget called the tensor product. In this video, we kick off a playlist on tensor products by giving a brief introduction to this object from the elementary co-ordinate viewpoint.

[wreynolds]

Great video! I admit that I was flummoxed by my first introduction to tensor products: it was in a commutative algebra course, in a general context of modules (over commutative rings), which themselves had only been introduced the week before, and the coordinate definition was not made explicit. Maybe this wouldn't have been so bad, except that none of the examples in the lectures used polynomial algebras; it was all Z and Z/n, meaning most of the time the tensor product was just zero. It took me a good while to fit things together for myself. So, I support the example-driven approach of this lecture!

[douglasholman]

This is tough stuff to wrap your head around, but I must say you make it possible to crack. Thank you professor for your generosity in providing these educational videos!

[just_another_guy]

Thanks for the lecture, looking forward to more videos on this subject.

[gucker]

Thank you for these lectures! They are very interesting and educative. I am so happy I found your channel!

[andreymelnik]

Great lecture! I enjoyed watching, thanks!
One issue is that it was not very clear why you brought up the direct sum in the very beginning.

[natefidalgo]

Another thing you may want to get into is the determinant of a nxn matrice it is a example of an n-linear object. As you where focusing on just bilinear (2-linear object). I say those examples are great for an intro. Also your coming at the theory of tensors from a modern/abstract/linear algebra point of view. There is another point of view on defining tensors its the differential forms /differential geometry /wedge/grassmainn/exterior calculus way.

[natefidalgo]

There is alot to tensors i am think of writing a program in c some day to do tensor arithmetic, calculus, and other things i find tensors get hard to compute as they get higher in rank so we should uses computers to double check us. We have many linear algebra libraries out there and math programs for linear algebra but far less tensor libraries that are popular, make it easy to understand, and easy to uses to date. Most people miss out on the multilinear algebra side of things and just get taught linear algebra. So they miss out on cliffords algebra, grassmainian, exterior/interior calculus, and alot of this is very important to properly progress the subjects of differential geometry as of current today...so its important to start teaching this stuff to progress the next generation.

[natefidalgo]

It might be worth noteing in the next videos that the independence of a bases is related to the Einsteins theory of relativity. And the fact is tensors are independent of the choice of a coordinate system. And you may want to get into covariant tensor, contravariant tensor, mixed tensors, the rank, order, properties and more arithmetic rules for them. If your going to show the connection from the abstract algebra tensor and the differential geometry tensors.