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Questions tagged [tensor-products]

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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General form of $\mathbb{Q}\otimes M$ as $\mathbb{Z}$-modules.

Compute the following tensor products (as $\mathbb{Z}$ modules): $\mathbb{Q}\otimes \mathbb{Z}/(n)$. $\mathbb{Q}\otimes\mathbb{Q}$. $\mathbb{Q}\otimes\mathbb{Q}/\mathbb{Z}$. $\mathbb{Q}/\mathbb{Z}\...
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Integration over cylinder surface of a fourth order tensor [on hold]

I would get the results of the following integration and the procedure of it. Thanks in advance.
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Associativity of mixed matrix product and tensor product?

Let $\bf A$ be an $l \times m$ matrix, $\bf B$ be an $m \times n$ matrix, and $\bf x$ be a row vector of any size. Will equality $\mathbf{A}(\mathbf{B}\otimes \mathbf{x})=\mathbf{AB}\otimes\mathbf{x}$ ...
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How can I find the component of this 3D tensor in a 2D space?

Good day Dear community, I'm really new in this field, so I truly appreciate your help and advises. I have this math problem: Consider the two mutually perpendicular unit vectors: $$i_a=3/5 i_1-4/...
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What do “projections” out of tensor products look like?

Convention. If $R$ is a relation $X \rightarrow Y$, what I mean is that $R$ is a subsets of $X \times Y$. We say that $X$ is the domain of $R$ and that $Y$ is the codomain. We'll write $x \overset{R}{\...
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Tensor product of $\mathbb{Q}$ and $\mathbb{R}$ [closed]

what is the result of the tensor product $\mathbb{Q}\otimes\mathbb{R}$? i.e. the elements form. Thank you
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Transformation Law for Tensor of Rank Two

Okay, I am sorry if this question seems absurd, but I am really having a difficult time understanding it. In the book by H. Jeffreys chapter Cartesian tensors, he defines a second order tensor as $ \...
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How to derive the non-torsion-free Bianchi identity by building a canonical torsion-free alternate derivative from the original covariant derivative

I know the Bianchi identity can be derived much more directly and simply (as is apparent in this post). The point here is to follow this alternate path to it as proposed in the paragraph after ...
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Generalizing the differintegral with fractional tensor rank

I wanted to generalize the derivative in a way that allows all differential (and integral) operators to be written using the same notation with as little modification as possible. What I came up with ...
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Problems on exercise 7.G in the book “K-Theory and C*-Algebras”

I have a lot problems on exercise 7.G in the book K-Theory and C*-Algebras by Wegge-Olsen. $\newcommand{\C}{\mathbb{C}}$ $X\subset \mathbb{C}$? As I know the character space of $C^*(u_1,u_2)$ is ...
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Physical dimensions in math

I was interested in the idea of of formalising the idea of physical dimensions with an algebraic structure containing "all physical quantities of any type". You'd need: Scalar multiplication over the ...
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equality of two simple tensors in $R=k[x,y]$

Consider $R=k[x,y]$, where $k$ is a field. Consider the $R$-module $M=\langle x,y\rangle$. I would like to see that $x \otimes y \neq y \otimes x$ in $M \otimes_R M$. My try to prove it: Let $F$ ...
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What are the commutative rings $R$ for which $A \otimes _{\Bbb Z} B = A \otimes _R B$ as abelian groups?

This is a follow up. What are the commutative rings $R$, for which given $R$-modules $A$ and $B$, $A \otimes _{\Bbb Z} B = A \otimes _R B$ as abelian groups? This is true when $R= \Bbb Q$, or $\...
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Is it true that $A \otimes_{\Bbb Z} B= A \otimes _{ R} B$ for every commutative ring $R$?

I have been obtaining a weird statement: Let $R$ be a commutative ring $A,B$ both $R$-modules. Then $A \otimes_{\Bbb Z} B= A \otimes _{ R} B$ as abelian groups. Proof: We may regard $A \otimes _{...
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Skew product of Hilbert Spaces

I’m researching into relations of Fock spaces (in particular so-called “exponential types”) and in the book “Introduction to Algebraic and Constructive Quantum Field Theory” by Segal, Baez and Zhou ...
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what does it mean to be symmetric for tensors and Kronecker delta symbols and help explain this answer to me

i understand how to change 2 tensors into Kronecker delta symbols but unsure how they managed to transform back to just one. If someone could add all the steps to get to the answer that would be ...
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Charater of induced representation

Suppose we have an induced representation of $\theta: H \to GL(W)$, we define the space $$ V := \mathbb{C}[G] \otimes_{\mathbb{C}[H]} W, $$ $V$ has as a basis $\{e_r \otimes_{\mathbb{C}[H]} w\}_{r \...
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is it possible to describe a set of matrix-vector products equivalently in terms of a 3D array and a matrix?

Is it possible to define the matrix-vector products \begin{align} A_i x_i = b_i; \ \forall i = 1,\cdots,N \ , \end{align} where $A_i \in \mathbb{R}^{M \times K}$, $x_i \in \mathbb{R}^{K \times 1}$, ...
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Tensor product terminology in category theory?

Say that I have any homomorphisms of commutative rings, $A \rightarrow B, A \rightarrow C.$ I recently read that $B \otimes_A C$ is the pushout of the morphisms in the category of commutative rings. ...
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Tensor product of finitely generated algebras

I am trying to compute tensor product of finitely generated algebras over a field. I was able to compute few special tensor products. Is there a general technique which allows one to compute the ...
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Developing a cross product of tensors within integrals

I read in a book the following unproven statement: $\int_{s} u\times A n \, ds = \int_v ( u\times \nabla\cdot A + \mathcal{E}: A^T ) \, dv$ with a: 1st order tensor, n: normal vector of s, A: 2nd ...
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tensor rank of an element in a tensor product

Let $V$ and $W$ be finite-dimensional vector spaces over $k$ with $\text{dim}(V)=n$ and $\text{dim}(W)=m$. How can I see that every element $t \in V \otimes_k W$ has tensor rank at most $\text{min}\{...
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Symmetric Algebra over Tensor Product

Let $k$ be a field. I am interested in the symmetric algebra functor $S : k \text{-vect} \rightarrow k \text{-alg}$ taking a $k$-vector space $V$ to the symmetric algebra $S(V)$ over $V$, which is a ...
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44 views

Tensor product of modules.

¡Hello! Please give me a hint for this problem. " Let $R$ a ring, $L$ a left ideal of $R$, $I$ right ideal of $R$. Show if $M$ are a left $R$-module, then exists a bijective function of commutative ...
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Are $\mathbb{C}[x]/x^2 \otimes \mathbb{C}[x]/x^2$ and $\mathbb{C}[x]/x^2$ isomorphic as $\mathbb{C}[x]/x^2$-modules?

Are $\frac{\mathbb{C}[x]}{x^2} \otimes \frac{\mathbb{C}[x]}{x^2}$ and $\frac{\mathbb{C}[x]}{x^2}$ isomorphic as $\frac{\mathbb{C}[x]}{x^2}$-modules? I believe that they are but it I know that there ...
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Finite Length Modules

Let $R$ be a ring. If $M, N$ and $L$ are $R$-modules, with, $\ell(M), \ell(N), \ell(L) < \infty,$ and $M \times N \cong M \times L,$ it is true that $N \cong L?$ The same occurs replacing $\times$...
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Why is the tensor product of two vector spaces a vector space?

We defined the tensor product of $v$ and $w$ to just mean a "symbol" $e_{vw}$, then considered the subspace spanned by all these symbols, and finally we quotient out by relations to make the tensor ...
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Known conditions to make $A \otimes B$ be pos.def., even if $A$ is not pos.def.?

Are there conditions that I should demand to be sure that $A \otimes B$ is positive definite, even when allowing $A$ not being positive definite, while $B$ is positive definite?
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Commuting Elements in Tensor Products of C*-Algebras

I am working on exercise 7.G in the book “K-Theory and C*-Algebras” by Wegge-Olsen. Let $A$ be some unital C*-algebra, $u$ a unitary in $M_n(A)$, and $u’$ a standard unitary (which is defined to be a ...
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Injective tensor product (Ryan 3.3)

Let $Q :Z \rightarrow Y$ be a quotient operator and let $I$ be the identity operator on X.Show that the tensor product operator $I \otimes Q : X \hat{{\otimes}}_\epsilon Z \rightarrow X \hat{{\otimes}}...
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What is a basis for the space of multilinear maps from $V_1 \times \dots \times V_k \to W$?

I know that a basis for the space $L(V_1, \dots, V_k; \mathbb R)$ is $$\{\varepsilon^{i_1} \otimes \cdots \otimes \varepsilon^{i_k}\mid 1 \le \varepsilon^{i_j} \le \dim(V_j)\}$$ where $\varepsilon^{...
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Are antilinear forms part of the tensor algebra of a $\mathbb{C}$-vector field?

Let $V$ be a finite-dimensional vector space over some field $K$, $V^*$ be its dual and $\mathcal{T}(V)$ be its tensor algebra. If $K$ is either $\mathbb{R}$ or $\mathbb{C}$, then every multilinear ...
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If P is a Projective R Module and Q is an Injective R module then $P \otimes_{R} Q$ is Injective

I am doing some basics on Protective, flat and Injective but I have no idea how to proceed for this one. Any help is appreciated!
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coproduct of noncommutative algebra and commutative algebras

I have read the book "Rings with generalized identities" and I understand that the free product of asociative unital algebras are the coproduct of them, but I can't understand why this reduces to the ...
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Which ones of these two shows that the tensor product operation on two multilinear functions is bilinear?

Suppose $v \in V_1 \times \cdots \times V_k$ and $w \in W_1\times \cdots \times W_l$. The tensor product of $F \in L(V_1 \times \cdots \times V_k, \mathbb R)$ and $G \in L(W_1 \times \cdots \times W_l,...
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about maximal ideal and tensor

x $\in$ ${\mathfrak{m}^t \cap I \otimes M}$ for all ${t}$ if x $\in$ ${\mathfrak{m}^t(I \otimes M)}$ for all ${t}$. I know that $m$ is maximal ideal
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Are all bivectors in three dimensions simple?

I want to show that all bivectors in three dimensions are simple. If I understand correctly, a bivector is simply an element from the two-fold exterior product $\bigwedge^2V$ of a vector space $V$, ...
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How to calculate the wedge product of differential forms with arbitrary coefficients

I need to calculate the wedge product between some differential forms of the type:   $\omega=P_1(x_1, ..., x_n)dx_1+\cdots+P_n(x_1, ..., x_n) dx_n$ and $d\omega$, i-e, $\omega\wedge d\omega$. where ...
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about torsion and flat module

Can someone explain me how (i) implies (ii) ? What is that lopping off term does ? And why at n=0 tensor is not exact?
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Definition of the tensor product of representations

I'm a bit confused about the following definition: Let $\rho_1:G \to Aut(V_1)$, $\rho: G \to Aut(V_2) $ be two representations of the same group $G$. Then a tensor product of representations is ...
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Tensor product property

Can someone confirm if the below statement is correct? $\textbf{a}(\textbf{b}\bullet\textbf{c})=(\textbf{a}\otimes\textbf{b})\textbf{c}$ = $(\textbf{b}\bullet\textbf{c})\textbf{a}$
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How to rewrite $\left[ \begin{array}{c|c} I_{r}\otimes (K e_1) \\ \vdots\\ I_{r}\otimes (K e_n) \end{array} \right]$?

I have the following matrix: $\left[ \begin{array}{c|c} I_{r}\otimes (K e_1) \\ \vdots\\ I_{r}\otimes (K e_n) \end{array} \right]$ $e_i$ is the i-th column of the identity matrix with the same ...
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How to determine the rank of a Khatri-Rao product of two matrices based on their each rank

As is known to all, the Khatri-Rao product is defined as $\mathbf{C}=\mathbf{A}\odot \mathbf{B}=\left[\begin{matrix}\mathbf{a}_1\otimes\mathbf{b}_1&\mathbf{a}_2\otimes\mathbf{b}_2&\cdots \...
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finding a projection on tensor spaces

I want to specify a projection from $V \otimes W$ to itself . How should I go about doing this ? And, can I charcterize all such projections that are possible ? Edit 1 : $V$ and $W$ are any two ...
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Vector Space Isomorphism related to Tensor Product

$V$ and $W$ are finite dimensional vector spaces over $k$. I need a basis free isomorphism between $V^*\otimes_{k} W^*$ and $Bil_{k}(V\times W,k)$. My attempt: We have a bilinear map $V^*\times W^*\...
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Deriving the properties of derivative operators

The difference in the action of $\bar{\triangledown_a}$ and $\triangledown_a$ on vector fields and all higher rank tensor fields is determined by $${\triangledown}_a \omega_b=\bar{\triangledown}...
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Additive group of tensor product of commutative $k$-algebras with units

Let $A_1, A_2$ denote two unital, commutative $k$ algebras, where $k$ is a field. Then by definition $A_i$ are rings with a bilinear operation on $k$. When we speak of the tensor product $A_1 \otimes ...
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Tensor product over another tensor product

Let $R$ be a ring and let $A$ and $B$ be $R$-algebras. Construct $A\otimes_R B$, the $R$-algebra given by tensor product. Let then $C$ and $D$ be two $A\otimes_RB$-algebras. Clearly $C$ and $D$ are ...
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30 views

Prove that $(M/IM)\otimes_{R/I} (N/IN) \cong (M \otimes_{R} N)/ I(M \otimes_{R} N)$

Show that $(M/IM)\otimes_{R/I} (N/IN) \cong (M \otimes_{R} N)/ I(M \otimes_{R} N)$ I tried to define a map $\phi : (M/IM)\times (N/IN) \to (M \otimes_{R} N)/ I(M \otimes_{R} N)$ by, $(\bar{m},\bar{n}...
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1answer
32 views

Prove that $R[X_1,\dots,X_n] \otimes R[Y_1,\dots,Y_m] \cong R[X_1,\dots,X_n,Y_1,\dots Y_m] $

Prove that $R[X_1,\dots,X_n] \otimes R[Y_1,\dots,Y_m] \cong R[X_1,\dots,X_n,Y_1,\dots Y_m] $ (as R-algebras) My attempt: Defined $\phi: R[X_1,\dots,X_n] \times R[Y_1,\dots,Y_m] \to R[X_1,\dots,X_n,...