At what point in the curriculum should the tensor product be introduced?

I remember my linear algebra teacher mentioning tensor products as an advanced topic that would be covered in upper level algebra coursework. During undergraduate abstract algebra, tensor products were not mentioned. Later, my graduate algebra professor told me that the tensor product was required background knowledge that I should have learned in linear algebra. Ultimately, I ended up learning the tensor product in a special topics course in the physics department.

Several of my classmates who had attended different undergraduate institutions reported similar experiences. From my limited polling, this seems to imply that the tensor product has proven difficult to position within a conventional mathematics curriculum.

At what point during a mathematics degree should the tensor product be introduced?

• I've worked it into linear algebra once or twice. However, I have a proof prerequisite for my Junior level linear course. Even so, honestly, as much as I love $\otimes$, it is a bit much in linear. In a second course of theoretical linear algebra or Algebra II it makes more sense. Personally, I saw them concretely represented in my differential geometry and manifold theory coursework. They could be covered any time after abstract algebra has been covered. I would wager the vast majority of undergrad programs in the US fail to cover this topic. Dec 19 '18 at 21:05

2 Answers

(This pertains to U.S. universities.) Times may have changed, but when I was in graduate school (several places, and yes I know this is unusual, but I mention it because I'm talking about more than one data point) this was a standard topic that showed up somewhere in the standard first year (2-semester) graduate algebra sequence (e.g. Lang, Hungerford, Jacobson, etc.), and the only way an undergraduate would see this in a formal class would be when taking such a class as an undergraduate. Of course, someone also taking upper level physics classes might see physics/engineering tensors in classical mechanics, continuum mechanics, fluid dynamics, electrodynamics, etc. (in my case it was in the form of dyads that appeared in the second semester of a 2-semester sequence using Symon), and someone taking an honors level advanced calculus course at one of the few universities that offered such courses (think Loomis/Sternberg, Nickerson/Spencer/Steenrod, or Fleming) would see tensor products, but in these cases (and probably others I could imagine) we're either not talking about formal tensor products in which notions such as "universal mapping property" arise or we're not talking about standard undergraduate material. I checked my undergraduate linear algebra text, Hoffman/Kunze, which was the text used for a fairly stiff undergraduate course in linear algebra that I took in 1978, and I see some discussion of tensor products in Sections 5.6: Multilinear Functions and 5.7: The Grassman Ring (pp. 166-180), but in looking over this now I definitely remember that we skipped these sections (further evidence is that I also have margin notes and homework problems marked in Section 5.4, then none until Chapters 6 and 7 and the beginning of Chapter 8).

• I had pretty much the same experience. I saw multilinear maps and tensors in concert with the construction of differential forms and of course in physics the beloved intertia tensor. But, the universal magical quotient formulation for me came much later. Once upon a time my brother wanted to put the universal property into the intro to proofs course. I wonder, after teaching for a decade, is it still his desire :) Dec 19 '18 at 21:12
• @James S. Cook: When I reread my answer a few hours later I realized that I kind of skipped over a "middle ground introduction" to tensors/tensor-products --- what's done in Hoffman/Kunze and differential forms stuff (e.g. Fleming's book; but probably not Nickerson/Spencer/Steenrod because that book is insanely abstract) --- when I gave the dichotomy of physics/engineering tensors and the universal mapping property definition of a tensor product, but I didn't feel up to rewriting parts of my answer. However, with your comment "formulation for me came much later", I thought I'd mention it. Dec 19 '18 at 22:49

In a math department (not physics department) I'd introduce it after some more familiar constructions (quotient groups, cyclic groups with a choice of generator,...) have already been described a second time by a universal mapping property. Tensor products will be tough the first time no matter what, so it would help if the idea of a construction based on a universal mapping property is already known to the class in some other contexts.