Tagged Questions

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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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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23 views

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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45 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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28 views

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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28 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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24 views

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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1answer
44 views

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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28 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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1answer
27 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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23 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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1answer
19 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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26 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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31 views

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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1answer
37 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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38 views

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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32 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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28 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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1answer
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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30 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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19 views

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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1answer
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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52 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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23 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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48 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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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
51 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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1answer
67 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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1answer
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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1answer
73 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
64 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 ...
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1answer
21 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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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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2answers
19 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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1answer
38 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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46 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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1answer
8 views

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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23 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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62 views

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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32 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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1answer
26 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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50 views

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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71 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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10 views

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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73 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} ...