Talk:Tensor product

Definition

In the definition from bases, it says:

The tensor product $V\otimes W$ of V and W is a vector space which has as a basis the set of all $v\otimes w$ with $v\in B_{V}$ and $w\in B_{W}.$

Out of the blue appears the tensorproduct $v\otimes w$, just the thing that should be defined. Madyno (talk) 12:33, 25 November 2022 (UTC)

One usage of the symbol $\otimes$ is for vector spaces; the other is for vectors. For basis vectors $v\in B_{V}$ and $w\in B_{W}$ the expression $v\otimes w$, which is a basis vector for $V\otimes W$, should be thought of as a primitive symbol that doesn't reduce to anything simpler. Elements of $V\otimes W$ are sums of such symbols with numerical coefficients. You can of course take the tensor product of non-basis vectors—the result will be written in terms of tensor products of basis vectors, as described in the paragraphs following the one containing the passage you quoted. I think it would be helpful to many readers if some elementary examples could be added to the article showing how the notation works and what it is useful for. Will Orrick (talk) 03:48, 26 November 2022 (UTC)

I think this is a plausible concern; for now I have tweaked the definition accordingly. 1234qwer1234qwer4 23:58, 16 March 2026 (UTC)

Tensor product of n vector spaces for n>2

The article defines tensor products of a finite nonnegative integer $n$ vector spaces for only the case $n=2$. However, the "General tensors" section uses tensor products for general $n$, so we need to define somewhere tensor products in general (finite nonnegative integer). Thatsme314 (talk) 19:31, 17 October 2023 (UTC)

The case n > 2 is defined in section § Associativity. D.Lazard (talk) 20:48, 17 October 2023 (UTC)

Good catch! Now, we just need to define it for $n=0$. Also, it might be nice to re-organize the article to emphasize the general definition. Thatsme314 (talk) 01:00, 18 October 2023 (UTC)

Linearly disjoint subsection

The Tensor product#Linearly disjoint section is for no apparent reason specialising to the case of complex vector spaces, when all sibling sections are over a general field $F$, and this definition likewise should work for a general field as well.

The meaning of $\mathbb {C} ^{mn}$ needs to be clarified. If the intent is $m\times n$ matrices (the formula given for $T$ would suggest so), then $\mathbb {C} ^{m\times n}$ is better. $\mathbb {C} ^{mn}$ is rather vectors with $mn$ elements, which would also work, but then you need to explain how you map the indices — straightforward if you've already grasped the concept of tensor product, but far from straightforward for those who don't. 130.243.94.123 (talk) 16:47, 16 April 2025 (UTC)

Since ⁠$m\times n=mn$⁠, your suggestion does not changes anything. However, the phrasing was awful, and I have inproved it. Note that your post suggests that the basis elements of ⁠$\mathbb {C} ^{mn}$⁠ are necessarily indexed by the first ⁠$mn$⁠ natural numbers. After my edit, it becomes clear that this is not the case here D.Lazard (talk) 17:54, 16 April 2025 (UTC)

I agree that the section must be rewritten over a general field ⁠$F$⁠. Indeed, the only usage of linear disjunction that I have ever encountered is the case of two finite field extensions of the rational numbers or of an algebraic number field. I have never heard of linear disjunction over ⁠$\mathbb {C}$⁠. D.Lazard (talk) 20:44, 16 April 2025 (UTC)

Tensor products of modules

The Tensor product#Tensor products of modules over a ring section is doing the non-commutative case twice: once starting with “More generally …”, and then (immediately after) again as the subsection “Tensor product of modules over a non-commutative ring”. 130.243.94.123 (talk) 16:53, 16 April 2025 (UTC)

I agree to either merge the two sections, or remove the noncommutative case from the first subsection. D.Lazard (talk) 17:13, 16 April 2025 (UTC)

Tensor rank

Tensor rank is currently treated at length at Tensor (intrinsic definition)#Tensor rank, but shouldn't it be here instead? (Arguably Tensor (intrinsic definition) should not exist in the first place, but that's a different story.) Sławomir Biały (talk) 07:28, 17 March 2026 (UTC)

That article is a bit weird. For instance, it's not even linked from this one and also starts with the hatnote "This article assumes an understanding of the tensor product of vector spaces without chosen bases", despite having "definition" in the title. It also covers a lot of the same topics as this article. On the other hand, the sections "Tensor products of modules over a ring" and "Tensor product of algebras" in this article should probably be moved into the "Other examples of tensor products" section and possibly shortened, if this article is supposed to primarily cover vector spaces. 1234qwer1234qwer4 18:04, 17 March 2026 (UTC)

There is also the Tensor article... which has a lot of overlapping content as well. 1234qwer1234qwer4 18:08, 17 March 2026 (UTC)

The article tensor seems like the right home for anything related to an "intrinsic definition". Sławomir Biały (talk) 18:57, 17 March 2026 (UTC)