Questions tagged [tensor-products]

For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.

Tensor products allow us to build "linear" objects from "multilinear" ones. It can refer to: basic ones from linear algebra/module theory, or more sophisticated versions from differential/algebraic geometry (bundles/sheaves), functional analysis (Hilbert/Banach/locally convex spaces), or in their purely abstract incarnation (monoidal categories). To make the intent clear, this tag should only be accompanied by relevant other tags specifying the context.

If $V_1$ and $V_2$ are vector spaces over some field $F$, then the tensor product of $V_1$ and $V_2$ is a vector space $V_1\otimes V_2$ for which there is a bilinear map $\beta\colon V_1\times V_2\longrightarrow V_1\otimes V_2$ with the following property: for each bilinear map $B$ from $V_1\times V_2$ into a vector space $W$, there is one and only one linear map $f_B\colon V_1\otimes V_2\longrightarrow W$ such that $B=f_B\circ\beta$. If $v_1\in V_1$ and $v_2\in V_2$, then $\beta(v_1,v_2)$ is usually denoted by $v_1\otimes v_2$.

If $V_1$ and $V_2$ are finite-dimensional, then $V_1\otimes V_2$ is also finite-dimensional and $\dim(V_1\otimes V_2)=\dim(V_1)\cdot\dim(V_2)$.

3946 questions
98
votes
7 answers

Mathematicians' Tensors vs. Physicists' Tensors

It seems, at times, that physicists and mathematicians mean different things when they say the word "tensor." From my perspective, when I say tensor, I mean "an element of a tensor product of vector spaces." For instance, here is a segment about…
msm
  • 2,574
  • 1
  • 10
  • 21
68
votes
4 answers

Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that the induced sequence $$A\otimes_R…
Klaus
  • 3,817
  • 2
  • 24
  • 37
68
votes
5 answers

Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$ Can you tell me if my proof is right: $\mathbb{Z}/m\mathbb{Z}$…
67
votes
2 answers

Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have gained quite some intuition for tensor products and…
67
votes
10 answers

Why is the tensor product important when we already have direct and semidirect products?

Can anyone explain me as to why Tensor Products are important, and what makes Mathematician's to define them in such a manner. We already have Direct Product, Semi-direct products, so after all why do we need Tensor Product? The Definition of Tensor…
anonymous
58
votes
3 answers

Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$?

I have a couple of questions about tensor products: Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$? Why is an element of $V^{*\otimes m}\otimes V^{\otimes n}$ the same thing as a multilinear map $V^m \to V^{\otimes n}$? What is the…
Eric Auld
  • 26,353
  • 9
  • 65
  • 174
52
votes
1 answer

Direct Sum vs. Direct Product vs. Tensor Product

There are a lot of questions like this all over the site, but I cannot find one that resolved my confusion- what are the formal definitions of direct sums, direct products, and tensor products (in the most general sense), and how are they different?
user247773
46
votes
1 answer

Was Grothendieck familiar with Stone's work on Boolean algebras?

In short, my question is: Was Grothendieck familiar with Stone's work on Boolean algebras? Background: In an answer to Pierre-Yves Gaillard's question Did Zariski really define the Zariski topology on the prime spectrum of a ring? I let myself get…
45
votes
6 answers

understanding of the "tensor product of vector spaces"

In Gowers's article "How to lose your fear of tensor products", he uses two ways to construct the tensor product of two vector spaces $V$ and $W$. The following are the two ways I understand: $V\otimes W:=\operatorname{span}\{[v,w]\mid v\in V,w\in…
user9464
41
votes
2 answers

Understanding isomorphic equivalences of tensor product

I get some big picture of tensor and tensor product by reading their Wikipedia articles, and several questions and answers posted before by others. But I cannot figure out how to show the following isomorphic equivalences: from Zach Conn: For…
Tim
  • 43,663
  • 43
  • 199
  • 459
40
votes
2 answers

Tensor product and Kronecker Product

Is there any difference between tensor product and Kronecker Product?
Manikanta Borah
  • 407
  • 1
  • 4
  • 4
40
votes
2 answers

Tensor products commute with direct limits

This is Exercise 2.20 in Atiyah-Macdonald. How can we prove that $\varinjlim (M_i \otimes N) \cong (\varinjlim M_i) \otimes N$ ? Atiyah-Macdonald give a suggestion, they say that one should obtain a map $g \colon (\varinjlim M_i )\times N…
Jr.
  • 3,826
  • 2
  • 27
  • 50
36
votes
6 answers

What, Exactly, Is a Tensor?

I've repeatedly read things that reference tensors, and despite reading the wiki page and other answers here on stackexchange I still don't know what a tensor is. I'm fine with hearing things in the framework on linear algebra, abstract algebra, and…
user82004
36
votes
4 answers

Tensor product algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$

I want to understand the tensor product $\mathbb C$-algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$. Of course it must be isomorphic to $\mathbb{C}\times\mathbb{C}.$ How can one construct an explicit isomorphism?
35
votes
3 answers

Symmetric and wedge product in algebra and differential geometry

Which is the correct identity? $dx \, dy = dx \otimes dy + dy \otimes dx$ $~~~$or$~~~$ $dx \, dy = \dfrac{dx \otimes dy + dy \otimes dx}{2}~$? $dx \wedge dy=dx \otimes dy - dy \otimes dx$ $~~~$or$~~~$ $dx \wedge dy=\dfrac{dx \otimes dy - dy \otimes…
1
2 3
99 100