Introduction to Tensor Products - some advice please

[Math Amateur]

I am (trying to :-) ) reading Dummit and Foote Section 10.4 on Tensor Products of Modules and am finding D&F's introduction to the topic of tensor products quite bewildering! ...

Can anyone give me a simple definition of a tensor product of modules together with an example to give me a basic understanding of the structure of tensor products of modules ... I would really appreciate such help ...

Alternatively, can anyone give me a reference that gives a good basic introduction to tensor products of modules?

... after looking at a number of texts I am getting the impression that maybe I need to go back and study bilinear forms. What do MHB members think ... do I need to do this?

Appreciate some help and advice ...

Peter

 

[Chris L T521]

I'm no pro at algebra, but I'm a fan of Keith Conrad and his notes (you can find the ones on tensor products here).

Hopefully this will make the topic of tensor products easier to understand (I, too, didn't like Dummit and Foote's explanations either)! (Smile)

 

[Math Amateur]

I'm no pro at algebra, but I'm a fan of Keith Conrad and his notes (you can find the ones on tensor products here).

Hopefully this will make the topic of tensor products easier to understand (I, too, didn't like Dummit and Foote's explanations either)! (Smile)

Thanks Chris, appreciate the help ...

Will give the notes a close look ...

Peter

 

[Math Amateur]

Thanks Chris, appreciate the help ...

Will give the notes a close look ...

Peter

Hi Chris,

Just glanced through the material on tensors ...

Wow! Thank you ... looks at first glance like a wonderful resource!

Will let you know how I go after some detailed study and reflection ...

Thanks so much!

Peter

 

[blue_raver22]

Hello Peter,

First of all, don't be discouraged by finding the introduction to tensor products of modules bewildering. It is a complex topic and it's completely normal to struggle with it at first. My advice would be to take your time and really try to understand the underlying concepts before moving on to more advanced material.

To put it simply, a tensor product of modules is a way of combining two modules to create a new module. It is denoted by the symbol ⊗ and is defined in terms of bilinear maps. A bilinear map is a function that takes two inputs and produces a scalar value, and it satisfies certain properties. In the context of modules, these inputs are elements from the two modules being multiplied.

For example, let's say we have two modules A and B. The tensor product A ⊗ B would be a new module that is created by taking all possible combinations of elements from A and B and multiplying them together. This is where the concept of bilinear maps comes in - the multiplication operation in the tensor product is defined in terms of a bilinear map.

A common example of a tensor product of modules is the cross product of two vector spaces. In this case, the two modules are the vector spaces and the tensor product creates a new vector space that is perpendicular to both of the original vector spaces.

As for references, I would recommend starting with a basic linear algebra textbook that covers tensor products of vector spaces. Once you have a solid understanding of that, you can move on to more advanced texts on tensor products of modules.

Whether or not you need to study bilinear forms depends on your level of understanding and the specific context in which you are studying tensor products. If you are struggling with the concept of bilinear maps, then it may be helpful to review bilinear forms. However, if you have a good understanding of bilinear maps, then you may not need to go back and study bilinear forms in depth.

I hope this helps clarify the concept of tensor products of modules for you. Don't hesitate to reach out for further clarification or assistance. Good luck with your studies!