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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Text introducing $T^{i,j}$-tensor algebra

I'm reading a lecture note here : http://www.cis.upenn.edu/~cis610/diffgeom7.pdf It introduces $T^{•,•}(M)$ the tensor algebra and says that this is a necessary tool in differential geometry. Well, ...
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Is the map $K\otimes_F Hom_F(V, U)\to Hom_K(K\otimes_F V,K\otimes_F U)$ surjective?

Given $ U, V$ vectorial space over F and $K/F$ a field extension ( even of infinite degree) the map $K\otimes_F Hom_F(V, U)\to Hom_K(K\otimes_F V,K\otimes_F U)$ is a monomorphism of K vectorial ...
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Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V):=B(V,V)$ are linear bounded endomorphisms with operator norm? If so, ...
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Inner product of tensor [closed]

If A is a tensor of type (1,2) and the inner product of A with some quantity B is a tensor of type (2,3) then B will be a tensor of type
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24 views

Free modules and their tensor product

can tensor product of two free non zero module over commutative ring with unity be zero? And can tensor product of two non zero vector spaces be zero space?
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33 views

Levi-Civita symbol (permutation tensor)

I was going over a past exam and the following two questions came up: Show that the Levi-Civita symbol $\varepsilon_{ijk}$ is a tensor. Evauluate the following: ...
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26 views

Linear algebra textbook for quantum computing?

I'm looking for an recommendation for a linear algebra textbook specifically to give me the background for learning about quantum computing, and quantum mechanics more generally. In particular, none ...
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23 views

General procedure to prove something is a tensor product of modules

I'm trying to understand some proofs of statements of the form: Show that some module is the tensor product of two other modules. When I'm looking at these proofs I always see that they start ...
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56 views

Express sum using tensor notation

So I have this expression $\sum_{l,m} c_{lmn} a_l a_m$. Where c is rank 3 tensor, and a is a rank 1 tensor. I would like to express this without specific reference to the tensor indices. Is there a ...
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1answer
25 views

Tensor derivative

What is the result of $$ \frac{\partial^2 \left(A^{ij}y^ix^j+B^{ij}x^iy^j\right)}{\partial \bf x\partial \bf y} $$ where $i,j$ obey Einstein summation convention, $A,B$ are constant, ${\bf ...
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1answer
31 views

Writing $e_1\otimes e_1+e_2\otimes e_2$ in another way [duplicate]

Let $V=\mathbb C^2$ with standard basis $\{e_1,e_2\}$. Are there $v,w\in V$ s.t. $e_1\otimes e_1+e_2\otimes e_2\in V\otimes V$ can be written as $v\otimes w$ Is the answer no ? for example if ...
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56 views

$\chi : \text{Hom}(M,A) \otimes \text{Hom}(A,N) \to \text{Hom}(M,N) $

Let $M,N$ be $A$-modules. Consider the map. $$\chi : \text{Hom}(M,A) \otimes \text{Hom}(A,N) \to \text{Hom}(M,N) $$ $$f \otimes g \mapsto g \circ f $$ If $M,N$ are projective I do know that ...
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1answer
52 views

Can we embed $X'\otimes Y$ into the space of bounded, linear operators $X\to Y$?

Let $\mathbb F\in\left\{\mathbb C,\mathbb R\right\}$ $X$ and $Y$ be normed $\mathbb F$-vector spaces $X'$ denote the topological dual space of $X$ $\mathfrak L(X,Y)$ denote the space of bounded, ...
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1answer
29 views

Are $X'\otimes Y$ and $\mathfrak L(X,Y)$ isomorphic?

Let $\mathbb F\in\left\{\mathbb C,\mathbb R\right\}$ $X$ and $Y$ be normed $\mathbb F$-vector spaces $X'$ be the topological dual space of $X$ $\mathfrak L(X,Y)$ be the set of bounded, linear ...
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1answer
37 views

An example of tensor product

Let $$ \otimes:R\times R\rightarrow W $$ $$ f:R\times R\rightarrow R~~,~f(X,Y)=XY $$ $\otimes$ is tensor product, $W$ is a vector space, and $f$ is a bilinear may. As I know , we need to find a ...
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82 views

The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category ...
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Minimal presentation of a tensor product of modules

Let $(R,m)$ be a local commutative noetherian ring. Suppose I have two modules $M$ and $N$ over $R$ given in terms of minimal presentations $$ M = \operatorname{coker}A, $$ and $$ N = ...
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41 views

Is $Z/mZ\otimes Z \cong Z/mZ$?

I'm reading a Homological Algebra book that states this in some point without proving. I was trying to prove it and it seems to me that the first module is infinite and the second is not.
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1answer
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stability of $K_0$ ($C^*$-algebras), question about the tensor product $K(H)\otimes A$.

I have a small question about stability of $K_0$: If $A$ is a $C^*$-algebra and $H$ is a separable infinite dimensional Hilbert space then $$K_0(A)\cong K_0(K(H)\otimes A),$$ where $K(H)$ denotes the ...
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29 views

Kernel of the map given by $j(m)=1\otimes m$ is in torsion module

Let $R$ be an integral domain, $K$ its field of fractions and $M$ an $R$-module. I want to show that the kernel of the map $j:M\rightarrow K\otimes M, m\mapsto 1\otimes m$ is contained in the torsion ...
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13 views

Linear independence in a tensor product space

Consider two N-dimensional vector spaces $V$ and $W$ over the same field and its tensor product $V\otimes W$. Let $\{v_i\}_{i=1}^N$ and $\{w_i\}_{i=1}^N$ be bases in $V$ and $W$. I want to show that ...
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36 views

An algebra isomorphism between $\mathbb{C}\otimes\mathbb{C}$ and $\mathbb{C}\oplus\mathbb{C}$

Considering $\mathbb{C}$ as an algebra over $\mathbb{R}$, it is easy to find a vector space isomorphism between $\mathbb{C}\otimes\mathbb{C}$ and $\mathbb{C}\oplus\mathbb{C}$. I am struggling to ...
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Representation decomposition over $GL_2(\mathbb{C})$

I have found that $Sym^2(V) \otimes Sym^2(V)$ can be decomposed over the special linear group as follows: $Sym^2(V) \otimes Sym^2(V) \simeq Sym^4(V) \oplus Sym^2(V) \oplus 1$ This is done using the ...
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12 views

CP decomposition representation for a tensor

Consider a third order tensor $X\in \mathbb{R}^{n_1\times n_2\times n_3}$, whose CP decomposition is $$X = \sum_{i=1}^r \sigma_i u_i \otimes v_i \otimes w_i, $$ where $\sigma_1\ge \sigma_2 \ge \cdots ...
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61 views

For $R$-modules M, is $M\cong R^{\oplus n}\otimes_RM\cong M^{\oplus n}$?

I wanted to explicitly give a bilinear map $R^{\oplus n}\times M\longrightarrow M^{\oplus n}$ when trying to prove that $R^{\oplus n}\otimes_RM\cong M^{\oplus n}$ and ended up with ...
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2answers
46 views

For all $R$-modules $N$, $R\otimes_R N\cong N$

Why is this so? This statement is from Aluffi's book Algebra: Chapter 0 and seemingly so trivial that it deserves no proof. So working with the universal property of tensor products, how exactly does ...
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2answers
30 views

Representing a linear transformation as a tensor

I understand that a linear transformation from a vector space $V$ to a vector space $W$ is a rank-$2$ tensor. What I would like some help with is how exactly to represent specific linear ...
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27 views

Does $\phi: A \otimes \mathbb{Q} \to B \otimes \mathbb{Q}$ surj. imply that for $b \in B$, $b = n \phi(a)$ for some $n \in \mathbb{Z}$, $a \in A$?

Let $A$ and $B$ be abelian groups. Suppose that we have a morphism $\phi: A \to B$ and that $\phi \otimes_\mathbb{Z} \mathbb{Q}: A \otimes_\mathbb{Z} \mathbb{Q} \to B \otimes_\mathbb{Z} \mathbb{Q}$ is ...
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92 views

$\mathbb{Z}[i]\otimes_{\mathbb{Z}}\mathbb{R}$ is isomorphic to the complex numbers

I am new to tensor poducts (of modules over a commutative ring with identity) and need to understand the following example to continue with the actual exercises in my material. Namely, I need to ...
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1answer
30 views

Prove the image of basis elements is linearly independent

I was wondering if someone could give me a quick proof or counterexample to the following statement. Let $f:V \rightarrow W$ be a linear map between finite dimensional vector spaces $V$ and $W$, both ...
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1answer
41 views

Components of the metric tensor and its inverse

Let $g$ denote a metric tensor. Then Wald writes (in his book on general relativity): "The inverse of $g_{ab}$...is a tensor of type $(2,0)$ and could be denoted as $(g^{-1})^{ab}$. It is convenient, ...
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1answer
36 views

Tensor Product of $\mathbb{Q}[\sqrt{2}]$.

How can one show that $\mathbb{Q}[\sqrt{2}] \otimes_{\mathbb{Q}[\sqrt{2}]} \mathbb{Q}[\sqrt{2}] \simeq \mathbb{Q}[\sqrt{2}]$ (which is a $2$ dimension vector space over $\mathbb{Q}$) and ...
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1answer
73 views

Confusion with abstract tensor notation

I am currently going through the abstract tensor notation in Wald's "General Relativity". I understand the purpose of it, but I need help understanding some of the conventions and definitions. So, ...
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28 views

why is $f\otimes g:A\otimes_{\min}C\to B \otimes_{\min} D$ injective?

If $f:A\to B$, $g:C\to D$ are injective $\ast$-homomorphisms between $C^*$-algebras $A, B, C, D$, is the induced map on the spatial tensor product $$f\otimes g:A\otimes_{\min}C\to B \otimes_{\min} ...
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1answer
39 views

Tensor product of dual vectors and vectors

I am reading "General Relativity" by Wald. At first he defines a tensor of type $(k,l)$ to be a multilinear map $T: V^* \times \cdots \times V^* \times V \times \cdots \times V \rightarrow ...
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1answer
27 views

Tensor product with vectors

I just started reading Wald's "General Relativity" and I am on his section regarding tensors. He defines the outer product as an operation on tensors of type of $(k,l)$ and $(k', l')$ which gives a ...
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1answer
26 views

Foiling two tensor products

I have this problem in exterior algebra where I have a function B and B is defined in the following ways. $$ B\left( \left( \begin{array}{c} u \\ v \\ w \end{array} \right), \left( \begin{array}{c} ...
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38 views

Irreducible components of tensor product representations.

Let $(\rho,V)$ be an irreducible representation of a finite group $G$, and let $W$ be a vector space. Then clearly $(\rho\otimes\text{Id}_{W},V\otimes W)$ is also a representation of $G$. I would like ...
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1answer
13 views

Dipole-Coupling Tensor: Electrostatic Dipole Moments

I've been struggling with this problem today. Here's an image of the question I'm attempting to answer. I'm relatively new to tensor algebra (I've been studying it for about a week or two), and I've ...
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32 views

Minimality of the Spatial C*-Norm

Given C*-Algebras $A,B$ and $x \in A \otimes_* B$ show that: (a) If $(\tau \otimes_* \rho)(x)=0$ for all $\tau \in A_+^*$ and $\rho \in B_+^*$, then $x=0$. PS: $\tau \in A_+^*$ means that $\tau$ is ...
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28 views

What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$?

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
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1answer
36 views

Basic question about the metric tensor

So I am reading Barrett O'Neil's Elementary Differential Geometry, which defines the metric tensor $g_p$ on a surface $M$ to be a bilinear symmetric positive-definite function on the set of ordered ...
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1answer
55 views

About the $1$ of ring

I could not find neither a proof nor a counterexample, can anyone solve this? Let $A$ be a finite dimensional $k$-algebra. (It not necessarily has $1$.) If $$\mu:A\otimes A \rightarrow A,\ ...
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2answers
30 views

Elementary tensors

Let $G,H$ be $R$-modules, and $G \otimes H$ be it's tensor product. I can't prove it and I suspect it's false that any element $\tau \in G \otimes H$ can be written as $\tau = g \otimes h$ for some ...
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2answers
38 views

$M \otimes_\mathbb{Z} N \cong \mathbb{Z}$ for cyclic modules $M,N$?

Is it true that $M \otimes_\mathbb{Z} N \cong \mathbb{Z}$ (considering $\mathbb{Z}$ as a $\mathbb{Z}$-module) if $M,N$ are cyclic $\mathbb{Z}$-modules (i. e. generated by one element)? I would ...
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Showing $R\otimes M \cong M$ for $R$-modules $R,M$

How to see that $R \otimes M \cong M$ if $R$ and $M$ are $R$-modules (with $R$ being a commutative ring with unity)? I thought about defining $f: R\otimes M \rightarrow M$ by $(r,m) \mapsto rm$. ...
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15 views

Rank of a locally free $\mathbb Z[G]$- module

Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$. Here $\mathbb Z_p[G]=\mathbb Z_p\otimes_\mathbb ...
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1answer
20 views

What are the main differences between Tensor Products of Vector Space vs Modules vs Algebras

What I understand so far: I understand the definitions of vector space, modules, algebras. I am also acquainted with basic properties of Tensor Product of Vector spaces, especially the "universal ...
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1answer
16 views

Query on Tensor Product of Quaternion Algebras

In this proof of isomorphism of tensor product of quaternion algebras I have two queries: 1) Why does one need to "check $xy=yx$ for all $x\in C$ and $y\in D$? 2) How is the product $CD$ defined? ...
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15 views

How to make $S\otimes_R M$ into a left $S[G]$-module

I have a basic question on tensor products which might have been asked before-sorry if this is the case. Let $G$ be a finite group and $R\subset S$ any (unital) ring extension. Let $M$ be a finitely ...