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 ...

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3
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2answers
175 views

What is the scalar product of tensors?

Given there a vector space $V$ with a scalar product $g(v_1,v_2)$ on it, what is the scalar product on, say, $V \otimes V^*$ ? According to Jeffrey Lee's "Manifolds and Differential Geometry" (see ...
0
votes
2answers
36 views

A question about tensor product over rings .

Let $A,B,C$ be three rings such that $f:A\to B$ and $g:A\to C$ are ring homomorphisms. How is $B\otimes_A C$ defined? I am especially worried about how $b\otimes_A tc$ is defined, where $t$ is a ...
1
vote
0answers
12 views

Show that the rank of A is a non-decreasing function of time

I need to show that the rank of the matrix A defined as: $A(t)=\int_0^{t} a(\tau)\otimes a(\tau) d\tau$ where: $a:\Re_{>0} \to \Re^n$ is a non-decreasing function of time... How should I start? ...
2
votes
1answer
41 views

Example of a ring $R$ such that $R\otimes_R R\not\simeq R$

I want to show the following statement: If a ring $R$ is commutative and $I,J\triangleleft R$, then $$ R/I\otimes_R R/J\simeq R/(I+J). $$ I can easily show this, assuming that $1\in R$. What can be a ...
0
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0answers
27 views

The effect of the Levi-Civita symbol on matrix elements

Suppose the matrix $O$ is orthogonal i.e. satisfies $$\tag{1} O^TO = 1 $$ and is also special $$\tag{2} \det O =1. $$ One can write equation $(2)$ as $$\tag{2'}\varepsilon^{i_1i_2\dots ...
4
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0answers
80 views

Alternative introduction to tensor products of vector spaces

One of the main obstacles in understanding the tensor product is that, unlike many other algebraic structures, you cannot really get hold of its element structure. This confuses many beginners. The ...
0
votes
1answer
39 views

A question regarding tensor products from Vakil's notes.

Vakil's notes have the following exercise: If $M$ is an $A$-module and $A\to B$ is a morphism of rings, give $B\otimes_A M$ the structure of a $B$-module. I don't understand how to do this. How ...
0
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0answers
21 views

When do basic elements of a tensor product of two modules correspond to a multilinear map?

My understanding is that in differential geometry one typically defines a 2-tensor on V (an $\mathbb{R}$ vector space) as a multilinear map from $V \times V \rightarrow \mathbb{R}$. The collection of ...
0
votes
1answer
25 views

Showing that an $(S,T)$-bimodule is a right $S^{op} \otimes_\mathbb{Z} T$-module

The questions I'm trying to answer is as follows (all rings are unital): Let $_{S}B_{T}$ be an $(S,T)$-bimodule, and let $R = S^{op} \otimes_\mathbb{Z} T$. Show that $B$ can be made into a right ...
0
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0answers
38 views

Why does infinite tensor product associated with some vectors in the operator algebras?

I notice that in the definition of infinite tensor product in the operator algebras, such as C*-algebras and W*-algebras, every component in the product is associated with a vector(or s state) and ...
-2
votes
1answer
60 views

How can I show that the tensor product of $\mathbb Z$ and $\mathbb R$ as $\mathbb Z$-modules is isomorphic to $\mathbb C$? [closed]

How can I show that $\mathbb Z\otimes_{\mathbb Z}\mathbb R\simeq\mathbb C$ as $\mathbb Z$-modules? I'm unable to come up with a solution as I'm quite new to Commutative Algebra. Can someone ...
2
votes
2answers
45 views

Kernel of a Linear Map on A Tensor Product

Suppose I have the linear maps $ l,k: V \otimes V \rightarrow V \otimes V$ defined by $ l( e_{i_1} \otimes e_{i_2} ) = e_{i_1} \otimes e_{i_2} + e_{i_2} \otimes e_{i_1}$ and $ k( e_{i_1} \otimes ...
0
votes
1answer
21 views

Question of Kronecker Product

The following is from a paper: Question: The G should be a (k,n) matrix. However, the kron(ei,ej) should be a (n×n,1) vector. So the resulting G should be a (n×n,1) vector. Is anything wrong or ...
0
votes
0answers
24 views

How to write an analogue to matrix-vector multiplation with an extra dimension in tensor notation

My background is severely lacking in tensor algebra, and after a few days of looking into tensors I am still not able to even formulate this question quite correctly; my apologies for that. I am aware ...
0
votes
2answers
57 views

$ker (T \otimes id_{Z})=ker(T)\otimes Z$

Does $x\otimes y=0 \implies x=0$ or $y=0$. I don't think so, since its equivalent to $B(x,y)=0$ form some bi-linear form. But my teacher said $ker (T \otimes id_{Z})=ker(T)\otimes Z$ where ...
2
votes
2answers
84 views

Direct sum decomposition of weight spaces and relation to Tensor products.

There are 3 parts to the question that I am trying to understand, and while it is not homework it seems instrumental in decomposition modules into weight spaces and their relation to tensor products. ...
1
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0answers
44 views

Reference about Tensor Product

do you know some reference about tensor product of modules, with all elementary properties are proved ?? I want something a bit more explicit than "Commutative Algebra" of Atiyah and Macdonald. ...
0
votes
1answer
68 views

Partial derivatives exist, but the function is not differentiable

It is well-known that a function $f:\mathbb{R}^n\to \mathbb{R}$ can have the property that it is differentiable along any line through the origin, but not even continuous at the origin. Can the same ...
0
votes
0answers
14 views

Basic tensor product related question

Show that $Hom_{K}(V_1\otimes V_2,W) \cong Bil_{K}(V_1\times V_2,W)$ My attempt: send $f\in f \circ \pi$ where $\pi:V_1\times V_2 \to V_2\otimes V_2$ a usual bilinear map, so now by universal ...
5
votes
1answer
122 views

The kernel of $R \to A \otimes_R B$ is nil

Let $R \to A$ and $R \to B$ be two homomorphisms of commutative rings whose kernels are nil (i.e. consist only of nilpotent elements). Then the kernel of $R \to A \otimes_R B$ is also nil. See ...
1
vote
1answer
27 views

Cartesian product distributes over second factor in tensor product?

I was thinking: a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ is nothing other than $m$ linear maps from $\mathbb{R}^n$ to $\mathbb{R}$. A linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ is also ...
0
votes
0answers
35 views

General expression of smooth sections of tensor bundles.

On page 317 of John Lee's Smooth Manifolds it's said that if $(x^i)$ are local coordinates on a smooth manifold $M$, then sections of the tensor bundle $T^kT^*M=\bigsqcup_{p\in M}T^k(T^*_pM)$ over a ...
13
votes
1answer
412 views

Symmetric and wedge product in algebra and differential geometry

I have been struggling with this issue for a while (and asked a similar question here), but still not found a satisfying answer. The question boils down to: which is the correct identity? $dx \, dy ...
2
votes
4answers
95 views

Showing $U\otimes (V\oplus W) \cong U\otimes V \oplus U\otimes W $

I have to show $U\otimes (V\oplus W) \cong U\otimes V \oplus U\otimes W $ where $U,V,W$ are $K-$vector spaces. One way to give a linear map from left to right is: $$u\otimes (v,w)\mapsto (u\otimes v, ...
0
votes
0answers
21 views

What is $M\otimes_\mathbf{Z}(N/P)\cong ?$

Let $A=\mathbf{Z}[G]$ where $G$ finite, and let $M,N,P$ be $A$-mods. Is there any particularly nice relationship for $M\otimes_\mathbf{Z}(N/P)$?
0
votes
1answer
18 views

Isomorphism of a torsion product and a quotioten of torsion product.

I have the following problem: If $A'$ a submodule of the right $R$-module $A$ and $B'$ a submodule of the left $R$-module $B$, then $A/A' \otimes B/B' \cong (A\otimes B)/C$ where $C$ is the ...
2
votes
1answer
37 views

Tensor product and its homomorphisms.

Given $f:A_R\rightarrow A'_R$, $g:B_R\rightarrow B'_R$ R-module homomorphism we can define $f\otimes g: A\otimes_R B\rightarrow A'\otimes_R B'$ such that $(f\otimes g)(a\otimes b)=f(a)\otimes g(b)$ ...
1
vote
0answers
35 views

Why does $\rho_{\mathbf{Z}^t\otimes P}=\rho_R$ imply the isomorphsim of $\mathbf{Z}^t\otimes P\cong R$?

so I have happened upon a thesis regarding the calculations of various $\mathbf{Z}D_6$ modules and their isomorphisms and came across a technique which is bothering me. Let me give an example. Let ...
0
votes
0answers
23 views

Injectivity and uniqueness of some maps from tensor products

Let $V_A$, $V_B$, $W$ be real vector spaces, let $V_A^*$, $V_B^*$, $W^*$ be their dual spaces. Let \begin{align*} &\phi: V_A \times V_B \rightarrow W \\ &\psi: V_A^* \times V_B^* \rightarrow ...
1
vote
1answer
20 views

Tensor product of $\mathbb{R}^d$ and $\mathbb{R}^s$ as abelian groups

It is well known (and easy to prove) that $\mathbb{R}^d\otimes_{\mathbb{R}} \mathbb{R}^s$ is isomorphic as a vector space to $\mathbb{R}^{sd}$. Now, I would like to know a simple description of the ...
3
votes
0answers
88 views

Tensor products over monoids : Element structure

Let $A$ be a (commutative) monoid. Let $M$ be a right $A$-set and let $N$ be a left $A$-set. Then we can construct the tensor product $M \otimes_A N$, which is a set (of even $A$-set when $A$ is ...
-1
votes
1answer
55 views

How to calculate scalar product of two gradients in indicial notation?

Does someone knows how calculate scalar product of two gradients and put the result in terms of nabla operator? . $(\vec\nabla{\gamma})\cdot(\vec\nabla{\gamma})$ = ?
9
votes
0answers
149 views

Minimal and characteristic polynomials on tensor product spaces

Given two finite-dimensional vector spaces $V$ and $W$ over a common field $k$ as well as linear transformations $\varphi \colon V \to V$ and $\psi \colon W \to W$, what (if anything) can one say in ...
0
votes
2answers
33 views

Scalar multiplication on tensor products

I am in the process of getting comfortable with tensor products (it's going very slowly). My question needs the following setup: We have a finite $K$-algebra $A$ and $L\supset K$ fields. We ...
0
votes
1answer
51 views

Does $V\otimes_k K\cong W\otimes_k K$ imply $V\cong W$?

Let $V$ and $W$ be two $k$-vector spaces of the same dimension and $K/k$ any field extension. If $V\otimes_k K\cong W\otimes_k K$ as $K$-vector spaces then are $V$ and $W$ already isomorphic over $k$? ...
2
votes
1answer
130 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
3
votes
1answer
42 views

On isomorphisms of tensors of certain type

I've got a question form Gille and Szamuely's "Central Simple Algebras' and it's about vector spaces equipped with tensors of certain types. Let $V$ be a $k$-vector space. For a field extension ...
4
votes
0answers
91 views

Consistency of different ways to define the tensor product

It seems my trouble with understanding tensors stems from the following statement: More specifically, the statement: Namely, given $B: V \times W \to U$ and $\xi: U \to \mathbb{R}$, $\xi ...
0
votes
1answer
64 views

Decomposition of Tensor into a Product of Tensors

I was working on a text that made use of the assumption that, for some rotation matrix $a_{ij}$, and some tensor $U$, the tensor under rotation is represented by $U'_{\alpha}=a_{\alpha i}U_i$. It made ...
1
vote
0answers
68 views

Tensor product of free modules over free algebra

Suppose $M$ and $N$ are modules over a (commutative, unital) ring $S$. Let $R$ be a subring of $S$ such that $S,M$ and $N$ are all free, finitely generated modules over $R$. Question: Under what ...
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0answers
35 views

efficient least squares A = BX+CXD (solving for matrix X)

I am interested in solving a least-squares solution of the form $$ \operatorname{argmin}_X \| A - BX - CXD \|_F^2 $$ for large (rank in hundreds to thousands) matrices $A,B,C,D,X$ I know this is ...
3
votes
1answer
124 views

Double dot product in Cylindrical Polar coordinates - Strain energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$ 2W = σ_{ij}ε_{ij} $$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
0
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0answers
82 views

Tensor product of arbitrary categories

I would like to consider a definition of tensor product in the category of (small, finite, whatever is needed) categories, analogous to the tensor product of vector spaces. I will first rewrite ...
2
votes
0answers
48 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
2
votes
1answer
62 views

Tensor products and isomorphic algebras

I found that $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathbb{C} \oplus\mathbb{C}$ and that $ \mathbb{H} \otimes_{ \mathbb{R}} \mathbb{C} \simeq M_2( \mathbb{C})$. Could anybody hint me how ...
1
vote
0answers
15 views

Expanding a product of matrices with tensor product and transpose

I'm trying to expand the following product of $\pm1$ matrices $H_1, H_2, K_1, K_2$: $(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 (H_1-H_2)\otimes K_2^T)(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 ...
1
vote
0answers
46 views

Tensor product with an L on top

Looking at the definition of the Kunneth morphism in SGA4, XVII, 5.4.1.4, there is the notation $$Rf_*K \overset{\mathbb{L}}{\boxtimes}_{\mathcal{A}_0} Rg_* L \rightarrow Rh_*(K ...
1
vote
0answers
101 views

Surjectivity implies injectivity of finitely generated modules, localization?

The following problem is canonical: Suppose $A$ is a commutative unitary ring, and $M$ is a finitely generated module over $A$. If an endomorphism $f\colon M\to M$ is surjective, then it's also ...
0
votes
1answer
31 views

Ring structure on tensor product of two $A$-modules

Let $A, B, C$ be a commutative rings. Suppose I have two ring homomorphisms, $\alpha : A \rightarrow B$ and $\beta : A \rightarrow C$. I am trying to show that $B \otimes_A C$ has a ring structure ...
3
votes
1answer
76 views

Tensor product of duals

Can we show $V^* \otimes W^* \simeq (V \otimes W)^*$, when $V$ or $W$ is finite-dimensional without referring to the basis? I can inject $V^* \otimes W^*$ into $(V \otimes W)^*$ using the obvious ...