Talk:Tensor product

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This page is poorly constructed

Well, maybe that's an overstatement, but it appears as though mathematicians have clogged this page up with soggy, pointless math jargon and made it irritatingly difficult to use as a reference for, you know, actually calculating things with tensor products. If all else fails, can somebody actually push the examples to the top as a quick reference and push the frivalous tedium to the bottom, or even better, package the frivalous tedium into a completely separate page? Or give me permission to do so.

Definitions confounded by construction

The definition used in this article badly confuses the construction of a tensor product with the definition of tensor product. For an excellent intrinsic definition of the tensor product see the one on PlanetMath for example. In the current state the definition mixes a method of building a tensor product with an abstract definition of the tensor product. — Preceding unsigned comment added by 139.48.54.241 (talk) 16:04, 24 August 2016 (UTC)[reply]

I partly agree that the focus of this article is not ideal. However, I do not think that the planetmath is really much better, with the over-reliance on the universal property. Is there no way to make something reasonably explicit that is also satisfactory as a definition (even an intuitive one)? Sławomir Biały (talk) 17:34, 24 August 2016 (UTC)[reply]

The "definition" also badly confuses 'definition' and 'motivation'. Mixing in motivational remarks into the text of a definition shows negligence for the aesthetics and rigor of precise mathematical language. — Preceding unsigned comment added by 93.207.197.239 (talk) 17:24, 6 September 2016 (UTC)[reply]

Can't we define the tensor product the same way polynomials are handled? Quoting from there

A polynomial is an expression that can be built from constants and symbols called indeterminates or variables by means of addition, multiplication and exponentiation to a non-negative integer power. Two such expressions that may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication are considered as defining the same polynomial.

If it were done like in this article, we'd have first built a free ring involving finitely-supported sequences of coefficients, and then built a big equivalence relation for "can be algebraically manipulated into one another", and then quotiented out by the equivalence class of 0. This is crazy, but it's what this article does for tensors! Patterning a definition after the above could give

An element of the tensor product $V \otimes W$ is an expression that can be built from vectors in $V$ and vectors in $W$ by vector addition, subtraction, scalar multiplication, and application of a formal variable $F$ representing a multilinear map whose domain is $V \times W$ . Two such expressions that may be transformed, one to the other, by applying the usual properties of linear algebra and multilinear maps are considered as defining the same element.

which should immediately be followed by a simple example IMO (ex: $2 F (v, w)$ , $F (v, w) + F (v, w)$ , $F (v + v, w)$ , and $F (v,2 w)$ are all elements of the tensor product and are equal).

This would easily segue into the universal property, because the map $~ h : V \otimes W \to Z$ is just substitution of $h$ for the formal variable F.

If we have to have a formalish definition (again, polynomial doesn't have one; even polynomial ring doesn't go to the extent we do here) couldn't it come after general motivational remarks like this? 64.92.17.6 (talk) 16:28, 13 May 2017 (UTC)[reply]

"generalises the outer product"?

The lead states:

"... the tensor product

V\otimes W

of two vector spaces V and W is itself a vector space, together with an operation of bilinear composition denoted by

\otimes

from ordered pairs in the Cartesian product

V\times W

into

V\otimes W

, in a way that generalizes the outer product."

Surely the outer product is the operation being referred to (confusingly also often called the tensor product)? There is no generalization of the outer product as an operation. —Quondum 16:08, 6 January 2017 (UTC)[reply]

I usually think of the outer product as defined for coordinate vectors in

\mathbb {R} ^{n}

, and the tensor product as the generalization for arbitrary pairs of vector spaces. Sławomir Biały (talk) 17:52, 6 January 2017 (UTC)[reply]

Ah, okay, makes sense. Just like the dot product is essentially defined on coordinate vectors (despite being used in other senses), in contrast to terms like inner product. Perhaps we should simply emphasize which of the two meanings of tensor product is meant when used, such as by referring to the tensor product of vectors (or tensors) or the tensor product operator when the bilinear operator is meant. The article Tensor product should then also clearly define and distinguish both in the lead (currently it simply avoids using the term tensor product for the operation, even though it refers to it). I've tweaked Outer product to be a little clearer in this sense by my understanding; feel free to change/revert. —Quondum 18:33, 6 January 2017 (UTC)[reply]

"Quick Sense"

I object to including this because the reader is left wondering what "subject to" means, and a definition for arbitrary modules is far less elementary than for vector spaces. By the way, header titles are not capitalized, per WP:MOSHEADER. @Gedt11: I also object to it on the grounds that anyone wanting a quick definition should get a rigorous one as a quotient module - other readers will need more introduction. Plus it's more confusing to readers because a tensor product is more than an abelian group. R has to act on it in a suitable way as well.--Jasper Deng (talk) 18:18, 18 December 2017 (UTC)[reply]

Reorganization Suggestions

The current definition of a tensor product is just the construction — instead of this, the definition should be given by the universal property, and then from the definition of a tensor product, a vector space representing it should be constructed. Check out definition 3.1 in http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf — Preceding unsigned comment added by Username6330 (talk • contribs) 02:05, 20 December 2017 (UTC)[reply]

@Username6330: So, the idea is that in Wikipedia we prefer to give a concrete down-to-earth definition first even if it is not correct for theoresits’ point of views. We in fact give the universal property def in tensor product of modules since the target audience of the latter needs to see the correct definition first. — Taku (talk) 00:51, 30 December 2017 (UTC)[reply]

(irony = on) Agreed. The idea of Wikipedia in many cases unfortunately is to first give a wrong description to have people get an incorrect idea and only then, when they understood the wrong definition to provide the correct one with the effect that they then no longer understand the correct concept (irony = off). It would be much better to give a motivation (and to make it clear that this is only a motivation) and then show how the precise definition evolves from that motivation. All good books do so. Just Wikipedia fails to do so in many places. :-/