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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Apparently meaningless computation with the Hodge star operator

In the last lecture we started speaking about hodge star operator. Let $E$ be a $n$ dimensional vector space with a non degenerate bilinear form $g$. $\mathcal{O}$ the orientation line of $E$, i.e. ...
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Algebraic operations with tensor products and direct sums.

When first learning about tensor products and direct sums of vector spaces, the analogy of multiplication and addition of vector spaces is sometimes used to help with the intuition. If we look at the ...
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Tensors, basic notation and components of

I'm trying to understand he basic notation(s) used to write out tensors, namely \begin{equation} T = T^{\nu_1,...,\nu_m}_{\mu_1,...,\mu_n} \frac{\partial}{\partial x^{\nu_1}} \otimes...\otimes ...
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23 views

Difference between to Tensor products with regards to modules

What would be the difference between $$ \otimes_B $$ and $$ \otimes $$ both in the following context and in general? Let A be a ring with $$ B \subset A $$ and M a B-Module. We can construct the ...
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31 views

Definition of Linear independence of algebraic $1$-forms

"Suppose that $a,b,c,d$ are linearly independent algebraic $1$-forms on $\mathbb{R}^n$". What does it mean for algebraic $1$-forms to be linearly independent? I have looked through my notes and ...
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Homotopy of morphisms of double complexes in Cartan Eilenberg

Let $s\colon f_1\cong f_2$ be a homotopy of two morphisms of complexes $A\to A'$ and $t\colon g_1\cong g_2$ be a homotopy of two morphisms of complexes $C\to C'$. I want to understand why ...
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34 views

Show that $x\otimes x+2\otimes2$ is not an elementary tensor in $I\otimes_{A}I$ [duplicate]

Let $A=\mathbb{Z}[x]$ and $I=(2,x)\lhd A.$ Show that $x\otimes x+2\otimes2$ is not an elementary tensor in $I\otimes_{A}I$. I have $$I\otimes_A I = \frac{L_A(I\times I)}{T}$$ where $T$ is the ...
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1answer
72 views

Is $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\bigotimes_{\mathbb{C}}\mathbb{C}$? [duplicate]

I'm trying to see if for several cases changing the ring in a tensor product affects the result or doesn't. Now I'm trying to prove $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq ...
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54 views

tensor product and commutation, category theoretical argument

It is a well-known fact that taking direct limits commutes with tensor products in the following sense: Let $I$ be a directed set and suppose for every $i\in I$ we have modules $M_i,N_i$ over a ring ...
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Can I reform this to a tensor/matrix product?

so I have the following vector matrix product: $$v = A w$$ Now I have this $n$-times: $$v^{(n)} = A^{(n)} w^{(n)} \quad \forall n$$ Is there any way to write this without $\forall$. Maybe somthing ...
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Calculate the tensor product of two vectors

Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis ...
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32 views

Isomorphism with tensor product

Let $f$ : $\Bbb Z_2$ $\rightarrow$ $\Bbb Z_4$ given as $f$($a$ + $2$$\Bbb Z$) = $2a$ + $\Bbb Z_4$ is a monomorphism . And knowing that <0,2> as subgroup of $\Bbb Z_4$ is isomorphic to $\Bbb Z_2$ ...
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23 views

Taking complex conjugate of an element in $\mathfrak{su}(2)_\mathbb{C}$

I read from a book that the complexification of the Lie algebra $\mathfrak{su}(2)$, noted $\mathfrak{su}(2)_\mathbb{C}$, is in fact the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$, the reason being: ...
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31 views

Multiplication in symmetric product space

STATEMENT: Let $V=\mathbb{R}^2$.Take $Y:=\left\{x\cdot y: x,y\in V\right\}$ where $S_2(V)$ is the symmetric product of $V$. QUESTION: What is multiplication in the symmetric product space?
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Is the map from representation ring to class functions a isomorphism?

I have a questions from representation theory. ($G$ is finite group) Fulton and Harris in "Representation Theory. A first Course" write that: the character defines a map $$\chi : R(G) \to ...
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1answer
20 views

When is the Tensor product of Modules itself a Module?

If $M$ is a right $R$ module and $N$ is a left $R$ module then $M \otimes_R N$ is an abelian group. Wikipedia says that if $M$ is an $R$ bimodule then $M \otimes_R N$ can take on the structure of a ...
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How is this expression well-defined?

I am going through the book "Introduction to Tensor Product of Banach Spaces" by Raymond Ryan. The tensor product of vector spaces is introduced in the first chapter which I briefly outline now. Let ...
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35 views

Understand the meaning of tensor product of modules

I am reading Atiyah's Introduction to Commutative Algebra. I have difficulty understanding the meaning of free A module $A^{(M\times N)}$. Here M and N are both A modules. In this book, the free A ...
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47 views

Tensor product of function space and vector space

In one book I found that they treat vector valued functions as tensor product of some function space and vector space. So for example $$ C(\mathbb{R},V) \simeq C(\mathbb{R})\otimes V \\ ...
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1answer
57 views

Decomposition of $\mathbb{Q}(\sqrt{2})\otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{3})$ and $\textbf{F}_q(t)\otimes_{\textbf{F}_q(t^q)}\textbf{F}_q(t)$

I have two questions about splitting of the tensor product into the product of fields How can one find a decomposition of $$\mathbb{Q}(\sqrt{2})\otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{2})$$ and ...
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evaluation of $\nabla \cdot ( \boldsymbol{B}(\boldsymbol{x}) \cdot \boldsymbol{B}^{T}(\boldsymbol{x}) )$

i have a symmetric positive matrix $\boldsymbol{D}(\boldsymbol{x})$ which can be decomposed as: $\boldsymbol{D}(\boldsymbol{x})$ = $\boldsymbol{B}(\boldsymbol{x}) ...
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29 views

The basis of this particular tensor product

I'm studying Fulton's Algebraic Curves book and he defines the module of differentials $\Omega$ in the following manner (R is a ring containing an algebraically closed field $k$): Let's define the ...
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Extension of scalars $M_B=B\otimes_A M$

Most textbooks say that the $B$-module structure on $M_B$ (for $A\rightarrow B$ a ring morphism and $M$ an $A$-module) is "defined" by $b'(b\otimes m)=b'b\otimes m$. How is this a proper definition? ...
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Other proof for tensor product equal to zero (without use Nakayama lemma and Jacobson radical)

Let $R$ is commutative local ring, $M$ is $R$-module, $N$ is $R$-module. If $M,N$ are finitely generated, how to prove: $M \otimes_R N=0$ if and only if $M=0$ or $N=0$. If we delete the ...
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33 views

Basis of a tensor

I'm new to tensors, and I need to understand what a certain basis actually is, how to visualise it. Say we have the $r$-dimensional vector space $T_p M$ and $n$-dimensional dual space $T^{*}_p M$. ...
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28 views

Complex conjugate and tensor product

Let $V$ be a real vector space and $f : V \rightarrow V$ a linear endomorphism. Also, let $\sigma : \mathbb{C} \rightarrow \mathbb{C}$ be complex conjugation. If $A$ is a real matrix, then it is ...
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24 views

Tensor product of the fraction field of a domain and a module over the domain

Given a fraction field $k(x)$ of the polynomial ring $k[x]$ over a field $k$ and an integral domain $R$ that is also a $k[x]$-module, is it true that $k(x) \otimes_{k[x]} R \cong Frac(R)$? I ...
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15 views

Tensor decomposition: How to determine the number of components (rank) of a CP decomposition?

I need to compute a best $rank-R$ CP decomposition of a tensor built from real world 3-dimensional data, of sizes approximately $100×200×300$. The best rank-R is determined by a given metric, which ...
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20 views

What are zeroes in $\mathbb{Q}\otimes M$

If regard $\mathbb{Q}$ as $\mathbb{Z}$-module, and M another $\mathbb{Z}$-module, then $q\otimes m$ is zero in $\mathbb{Q}\otimes M$ iff q or m is zero, how to proof it?
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27 views

Functorial isomorphism involving tensor products

Let $R$ be a commutative ring and $E', E, F', F$ be free, f.g. $R$-modules of equal rank. For $f\in L(E',E):={\rm Hom}_R(E',E)$ and $g\in L(F',F)$, let $T(f.g)\in L(E'\otimes_R F', E\otimes_R F)$ be ...
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Universal property of the Tensor Algebra

Let M be an A-module over a commutative ring A. For any A-algebra N and A-module homomorphism $\phi : M \rightarrow N$ there is a unique A-algebra homomorphism $\Phi : T(M) \rightarrow N$ (where T(M) ...
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38 views

Simple question on tensoring by a quotient ring

$A \subset B$ is an extension of commutative rings s.t. $B$ is a f.g. free $A$-module of rank $n$, so I have $A^n \stackrel{\sim}{\longrightarrow} B$ as $A$-modules. Let $\mathfrak a$ be an ideal of ...
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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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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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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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22 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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33 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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41 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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55 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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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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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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1answer
31 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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1answer
55 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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74 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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8 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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74 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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1answer
71 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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1answer
28 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 ...