# Tensor Product of Vector Spaces - Math (Algebraic) Reference

I am quite sure that this question will be marked as duplicate and I am very well aware of all the other threads on the topic, which however do not provide me with a satisfactory answer.

Essentially what I am asking for is a good reference on tensor products of vector space. By reference I mean a pure math reference, nothing like physics/engineering/handwaving.

My background on relevant topics: I think I have a decent understanding of abstract linear algebra (more or less at the level of Axler - Linear Algebra Done Right or Hoffman & Kunze - Linear Algebra - at least first 9 chapters) but I had little exposure to Abstract Algebra (just a little about groups, very very little about rings and fields and zero exposure to modules).

I spent several hours in the library and on Amazon but I surprisingly found very little math material, probably only Greub - Multilinear Algebra, which is maybe a little too heavy for me (but probably as of now would be my best shot).

Also, I am aware of K. Conrad's expository papers "Tensor products I" and "Tensor products II" and, from what I can understand by having a look at them, they would be perfect for me if they were written just in a vector space setting. However the module setting seems a bit too hard for me: I dont even know what a module is but I think I nonetheless have the right to understand what a tensor product of vector spaces is.

If your background includes a reasonable understanding of something like Linear Algebra Done Right, I think there isn't anything to fear from approaching Greub's Multilinear Algebra. The prerequisites are actually what is treated in Greub's Linear Algebra and not much else.

I would also recommend chapter 14 of Roman's Advanced Linear Algebra (see https://www.amazon.ca/Advanced-Linear-Algebra-Steven-Roman/dp/0387728287/ref=sr_1_1?keywords=roman+advanced+linear&qid=1569758068&s=gateway&sr=8-1). It does a reasonably good job of explaining tensor products and doesn't require anything beyond prior exposure to Linear Algebra and perhaps some basic Abstract Algebra.

Hope this helps!

• I had the impression Roman was a bit too heavy on Abstract Algebra. See his chapters 4, 5 and 6 where he goes quite in depth on modules. I barely know the definition of a module.. Did you read the book? Do you know if I can skip something and still make it into chapter 14?
– Tom
Commented Sep 30, 2019 at 8:48
• @Tom I did read Roman's text and the modules bit isn't needed at all. You can jump right into 14 (it's what I did from just a linear algebra and basic knowledge of groups and rings etc and even those bits of abstract algebra aren't very essential in my opinion). Commented Oct 3, 2019 at 13:45
• Very interesting - Thanks a lot - I ll try!
– Tom
Commented Oct 3, 2019 at 14:10