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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Tensor products isomorphic to hom-sets with a structure

In which cases the tensor product of objects, say A and B, is (isomorphic with) the objects with the carrier set Hom(A,B) and a corresponding structure?
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35 views

using a symmetric matrix to simplify expression involving antisymmetric matrices

I am trying to simplify the following expression --$$(\vec{A} \times \vec{B})\cdot\underline{\underline{S}}\cdot(\vec{C} \times \vec{D}) = (A_jB_k\epsilon_{jki})S_{im}(\epsilon_{mnp}C_nD_p)$$ $S$ is a ...
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31 views

identify a tensor product by virtue of pure and entangled elements

If I take a tensor product of vector spaces (for simplicity - this could be more general) $V\otimes W$ then of course it is a vector space, but it has additional structure. One way to think about ...
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26 views

Prove a tensor product's component is zero

Suppose we have $R$-module elements $m\in M,n\in N$ with $m\otimes n=0\in M\underset{R}\otimes N$. Is it necessarily true that $m=0$ or $n=0$? I can't seem to prove it using the basic tensor product ...
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45 views

Tensor product of linear transformations

If $U$ and $V$ are finite-dimensional vector spaces then $U^*\otimes V^* \approx (U \otimes V)^*$ via the isomorphism $\tau: U^*\otimes V^* \to(U \otimes V)^*$ given by $\tau(f \otimes g)(u \otimes v) ...
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When is the rank of a general tensor 1?

Suppose we have two modules $M$,$N$ with $\mathbf{Z}$-bases, and take the tensor product $M\otimes_\mathbf{Z} N$. If I use Bourbaki's definition that the rank of a general tensor T is defined to be ...
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43 views

Can you always construct a map $A\otimes B\to A\times B$?

Suppose we have two $R$-mods $A,\,B$. Can we always construct a homomorphism $A\otimes B\to A\times B$?
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64 views

How to identifiy $V \wedge V$ with the space of all alternating bilinear forms

Let $\{ e_i \}$ be a basis for $V$, then the space of tensors $V \otimes V$ could be identified with the space of all formal sums $\sum_{ij} \alpha_{ij} (e_i, e_j)$ (I know a base independent approach ...
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29 views

Tensor product of two linear map and its matrix representation

Suppose $T_1: \mathbb{R}^n\to\mathbb{R}^n$ be any linear map and wrt a basis $\{e_1,\dots,e_n\}$ the matrix of $T_1$ is $M$, aand $T_2:\mathbb{R}^m\to\mathbb{R}^m$ be another linear map whose matrix ...
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53 views

Unnecessary Elements in the Tensor Product?

For vector spaces $U, V$ there exits a unique (up to isomorphism) vector space, denoted by $U \otimes V$, and a bilinear map $\eta : U \times V \to U \otimes V$ such that for every bilinear map $\xi : ...
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35 views

Isomorphism between compact operators and compact operators tensor matrices ($\mathbb{K}\otimes M_n(\mathbb{C})\cong \mathbb{K}$)

Let $\mathbb{K}$ be the compact operators and $M_n(\mathbb{C})$ the complex valued matrices. I have read the algebra $\mathbb{K}\otimes M_n(\mathbb{C})$ is isomorphic to $\mathbb{K}$. Could you tell ...
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66 views

Proof that the tensor product is the coproduct in the category of R-algebras

Given the category of commutative R- or k-Algebras, it is often mentioned that the coproduct is the same as the tensor product. I'm interested in the proof of this statement. One idea would be to ...
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91 views

Different Definitions of Tensor product, Halmos, Formal Sums, Universal Property

In the classic Finite-Dimensional Vector Spaces by P. Halmos he defines the Tensor product as The tensor product $U \otimes V$ of two finite-dimensional vector spaces $U$ and $V$ (over the same ...
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dim of $\Bbb R^3 \otimes_\Bbb R \Bbb C$ when considering as a $\Bbb C$-vector space

I'm looking at Sergei Winitzki's Linear Algebra via Exterior Products, and he has a question on tensor products. Firstly we construct the real vector space $\Bbb R^3 \otimes_\Bbb R \Bbb C$ which is ...
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10 views

continuity on the common edge of the two tensor-product Bezier surfaces

The two tensor-product Bezier surfaces,with control points cij and dij What are the conditions on the control points that ensure that p and q join with c^1 continuity on the common edge s = 1, 0 ≤ ...
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92 views

Rank of tensors in terms of ranks of associated linear maps

Let $V$ be a vector space over a field $k$, let $w \in \otimes^l V$ be a tensors. We call $w$ a simple tensor if it can be written as $$ w=w_1 \otimes w_2 \otimes \ldots \otimes w_l, $$ where $w_i \in ...
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78 views

Show structure of a commutative ring in a tensor product [closed]

I need some help with this: Let $R$ be a commutative ring and $S$ and $T$ be commutative $R$-algebras. Show that $$ S \otimes T $$ has the structure of a commutative ring with multiplication: $$ (s ...
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30 views

How can I show that $(a^\top \otimes bb^\top \otimes a) = (a \otimes b)(b \otimes a)^\top$?

Let symbol $\otimes$ denotes the Kronecker product, $a \in \mathbb R^n$ and $b \in \mathbb R^m$. How can I show that $(a^\top \otimes bb^\top \otimes a) = (ba^\top \otimes ab^\top)$ ? My final goal ...
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36 views

Under what conditions can I expect the restriction of scalars functor to preserve tensor products

Suppose I have the canonical injection $i:H\hookrightarrow G$. Evidently I can induce the map on modules which restricts scalars from $\mathbf{Z}[G]$ to $\mathbf{Z}[H]$; that is, ...
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35 views

What is the tensor product of Z/10Z with Z/12Z? [duplicate]

I've just met tensor products of modules in part of my self-study and as a concrete exercise I've been asked to calculate $\mathbb{Z}/(10){\otimes}_{\mathbb{Z}} \mathbb{Z}/(12)$. Short of writing out ...
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110 views

Outer Product of Two Matrices?

How would I go about calculating the outer product of two matrices of 2 dimensions each? From what I can find, outer product seems to be the product of two vectors, $u$ and the transpose of another ...
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48 views

Definition of Tensor Product of Modules

I am really struggling to understand several parts of the definition of tensor product given in my lecture notes: Definition of the tensor product *Denote by L the free A-module with a basis ...
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48 views

Is $R\otimes_R M\cong M$ when $R$ not necessarily commutative?

Suppose $R$ is not commutative. I am guessing in general that $R\otimes_R M\cong M$ fails to be true. However, are there any cases where this is still true? For instance group rings for a ...
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Tensor product of fields and its subalgebra

In Nathan Jacobson's Basic Algebra II, in section 8.18: Tensor product of fields he is discussing what happens to $E \otimes_FK$, when $K|F$ and $E|F$, and E is algebraic over F. At one point he ...
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28 views

Group action on a tensor product

Let $R \subset S$ be an extension of commutative rings, $G$ a group and $M$ a left $R[G]$- module. Then how do I make the tensor product $S\otimes_R M$ into a left $S[G]$- module? What is the action ...
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Using the universal property of tensor product to show that $(\Bbb{Z}/4\Bbb{Z}) \otimes (\Bbb{Z}/6\Bbb{Z})\cong \Bbb{Z}/2\Bbb{Z}$ [duplicate]

In the algebra lecture i need to solve the following exercise Use the universal property of the tensor product to show that $$(\Bbb{Z}/4\Bbb{Z}) \otimes (\Bbb{Z}/6\Bbb{Z})\cong ...
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35 views

Tensor products, help with proof

Let $X$, $Y$ and $Z$ be Banach spaces. Let the space $X\otimes_{\epsilon}Y\otimes_{\epsilon}Z^{*}$ be the injective tensor product. The injective norm is defined as follows: ...
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27 views

Then, how can we show that $\forall i,j\in \mathbb Z, s.t.:1\leq i<j\leq n$, $e_i\wedge e_j $ is a basis vector for $\wedge^2(V) $?

Let $V$ be a n dimensional vector space. Suppose $x,y\in V, f,g\in V^*$. Define $f\wedge g(x,y) = det \left( {\begin{array}{cc} fx & fy \\ gx & gy \\ \end{array} } \right)$ Then, how can we ...
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Augmentation ideal of tensor product of group rings

We have $\epsilon: \mathbb{Z}G\to\mathbb{Z}$ the augmentation such that $\epsilon(\sum z_gg)=\sum z_g$ and $\ker(\epsilon)=\operatorname{Aug}\mathbb{Z}G$. We have to $\mathbb{Z}(A\times A_1)\simeq ...
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79 views

Comparison of Hilbert space tensor product and wedge product

For Hilbert Spaces: $$(|0\rangle + |1\rangle)\otimes (|0\rangle + |1\rangle) = |00\rangle + |01\rangle + |10\rangle + |11\rangle.$$ where all results are column vectors \begin{eqnarray*} 0 ...
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dimension of the set of separable states is $\dim \cal H_1 + \dim \cal H_2$?

Please can you help me to understand how the dimension of the set of separable states is $\dim \cal H_1 + \dim \cal H_2$? This is the relevant passage: So far, we have assumed implicitly that the ...
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78 views

A question involving the module of differentials

Let $B$ be a local ring. Let $k$ be its residue field. Do we need $B$ to contain a copy of $k$ in order for the following to be true: $$\operatorname{Hom}_{k}({\Omega_{B/k}\otimes_{B} ...
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22 views

Smoothly Parametrized basis of $V\otimes V$

$\{(\cos t,\sin t),(-\sin t,\cos t)\}$ is a basis of $\mathbb{R}\oplus \mathbb{R}$. Is it also a basis of $\mathbb{R}\otimes \mathbb{R}$? If not, what's an easy way to construct a non-trivial basis ...
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63 views

Two different definitions of sheaf of $K$-modules and tensor products.

I am confused by two different approaches to defining sheafs of modules. In Hartshorne there is the concept of a sheaf $F$ of modules $O_X$-modules, where $F(U)$ is a module over $O_X(U)$ with ...
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70 views

Tensor product of abelian groups

Let $A$ and $B$ be abelian groups. For each $m>0$, show that $A\otimes Z_m \cong A/mA$. And describe $A\otimes B$, when $A$ and $B$ are finitely generated. ...
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1answer
82 views

Tensoring $k[x] \otimes_k k[y]$

Recalling result of tensor product of polynomial rings and $k[x]\otimes_k k[x]\cong k[x,y]$? suggest that $k[x] \otimes_k k[y] = k[x,y]$. I am sure that this is true if the tensor product is being ...
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42 views

Rank of abelian groups

I have read that given a $\mathbb{Z}$-module $M$, the maximal number of $\mathbb{Z}$-linear independent elements is given by $\operatorname{rank}M=\dim_\mathbb{Q}(\mathbb{Q}\otimes_\mathbb{Z}M)$. ...
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34 views

Tensor similarity measure

In some practical applications, such as in diffusion tensor imaging (DTI), the diffusion data is often represented by a symmetric positive definite second order tensor (basically a 3x3 matrix). The ...
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1answer
49 views

How to turn a tensor product into a matrix product?

I would like to do an operation on a matrix acting on a tensor product vector space that is a bit unusual. It is similar to a partial trace, but not quite that. Say I have a tensor product vector ...
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1answer
48 views

Is there an element-free proof that $\mathbb{Z}/(m)\otimes_\mathbb{Z}\mathbb{Z}/(n)\cong\mathbb{Z}/(m,n)$?

I'm aware of the proof which goes about showing the tensor product is generated by $1\otimes 1$, which has order $\gcd(m,n)$. Out of curiosity, is there an element-free proof?
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How do I take the inner product of these two tensors: $T^{ij}$ and $T_{ij}$

The tensors are of contravariant and covariant order two, respectively. Our teacher said something about the result being identity, or the kroneker delta $\delta_i^j$, I think, but I'm not too sure. ...
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Tensor Product, Exterior Power and Splitting

Let $M$ be a $\mathbb{Z}$-module and consider the submodule $K=\langle m\otimes m\mid m\in M\rangle$ of $M\otimes M$. Under what conditions does the SES $$0\to K\to M\otimes M\to M\wedge M\to 0$$ ...
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180 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 ...
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37 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 ...
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13 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? ...
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42 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 ...
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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 ...
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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 ...
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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 ...
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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 ...