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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Tensorial product, a simple question

I need to find the components of $D$: $$D=a\otimes a$$ where $a$ is a tensor of order 2. Thanks!
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29 views

Product of two geometric series

I have used the Product of two power series and find out the below results. But it is to some extend strange for me, could you please confirm the results? Let $A=\sum_{i=0}^{\infty}(\frac{L}{a})^i$ ...
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38 views

Tensor Product over a ring

Given Two Fields $F,K$, and two vector spaces $V,W$ over $F$, what does tensor product $$V\otimes_{K} W$$ mean? I am not certain whether this is defined in general. I came across it in cases wheh ...
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30 views

Tensor product of two simple modules

Let $k$ be a field of arbitrary characteristic. Suppose that $A$ and $B$ are finite dimensional $k$-algebras. Let $S$ be a finite dimensional simple $A$-module and let $T$ be a finite dimensional ...
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24 views

Another matrix of a given operator $A \otimes A$

Let $V$ be a $4$-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $(e_{1}, ...
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32 views

About the definition of the tensor product of modules

I was reading something about tensor product (balanced product) of modules (over an arbitrary ring $R$), but I cannot realize why we need a left and a right module. Would it be the same with two right ...
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25 views

What is a wedge product?

What is actually a wedge product ? How does it differ from a tensor product ? What is the intuition to invoke such a product, kind of like tensor product is invoked to simplify multilinear maps ?
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97 views

$k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$

This is part of an exercise from Eisenbud: $k$ is a field, describe as explicitly as possible a) $k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$ b) $k[x] \otimes_{k} k[y]$ Any hint ?
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53 views

Tensor product is o?

$K$ generated by $(ar,rb)$ for every $a\in A$, $b\in B$, $r\in R$. $F$ generated by $r(a,b)$ for every $a\in A$, $b\in B$, $r\in R$. $(a,rb)=(ar,b)=r(a,b)$,then $K=F$, $F/K=0$ Apparently I ...
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26 views

Tensoring and retaining projectiveness

Let $A$be a unital associative ring, If $A$ is not a projective $A$-bimodule, however $A\otimes A$ is a projective $A$-bimodule, can we conclude that $A\otimes A \otimes A$ is also projective?
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25 views

Matrix of a given operator $A \otimes A$

Let $V$ be a 3-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $e_{1}$, $e_{2}$, $e_{3}$ ...
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1answer
22 views

Are the generators of the subgroup defining tensor products linearly independent over $\mathbb Z$?

Let $S$ be a (commutative) ring with identity, and let $M$, $N$ be $S$-modules. (I guess if $S$ isn't commutative, I want $M$ to be a right $S$-module an $N$ a left $S$-module.) In the definition of ...
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8 views

Tensor product - result as column vector or matrix

I have a bit of quantum computing on my current semester and I am currently reading about tensors. I have read article on outer product (which is described as tensor product of two vectors): ...
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75 views

Tensor product of quotient rings [duplicate]

$A$ is a commutative ring with unit and $\mathfrak a$, $\mathfrak b$ ideals. I have to show that $$A/\mathfrak a \otimes_{A} A/\mathfrak b \cong A/(\mathfrak{a+b}).$$ Any hint ?
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10 views

Is the tensor formed by the tensor product of two positive semi-definite symmetric tensors itself positive semi-definite?

Let $A_{ij}$ and $B_{ij}$ be positive semi-definite and symmetric tensors. My question is then if we form a new rank-4 tensor $C_{ijkl}=A_{ij}B_{kl}$, does $C_{ijkl}$ also inherit the positive ...
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23 views

How is the multiplication between a multidimensional tensor with a matrix defined?

I am thinking this calculation in the following way but I am wondering if it is correct. Can anybody explain to me please? For example, I have a 3-way tensor $T^{u×i×t}$. How do I multiply this ...
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1answer
35 views

Example of $\sum_i a_i\otimes b_i\in M\otimes_AN$ which cannot be written as $a\otimes b$

In the appendix of my commutative algebra text: Note that in general the element of $M\otimes_AN$ is a sum of the form $\sum_i a_i\otimes b_i$ and cannot be necessarily written as $a\otimes b$. ...
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1answer
29 views

Relation between Tensor Product and Homomorphism

We know that there is a natural isomorphsm between $$V^*\otimes W \text{ and } Hom(V,W)$$ whenever either $V$ or $W$ is finite dimensional. (We also know that there always exists a linear map from ...
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107 views

If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
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1answer
31 views

Trace of tensor product vs Tensor contraction

I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality: $$ \text{Tr}(T_1\otimes ...
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1answer
36 views

submanifold of Euclidean space is oriented if and only if normal bundle is an oriented vector bundle.

Let $f:M\longrightarrow \mathbb{R}^{n+k}$ be an immersion of $n$-dimensional manifold $M$ into $\mathbb{R}^n$. Let $\nu(M)$ be the normal bundle of $M$. Prove that $M$ is oriented if and only if ...
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2answers
70 views

Tensor powers of injective linear maps of free modules

This is a basic question on tensor products of linear maps. Let $R$ be a commutative ring and let $\varphi: M\to N$ be an injective linear map of finitely generated free $R$-modules. Question: Are ...
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1answer
30 views

The space of all bounded sequences over a Banach Algebra.

If $A$ is a commutative Banach algebra, can we identify $l^\infty(A)$ as the dual of $ l_1\hat\otimes A$ (the projective tensor product of $l_1$ with $A$? Also, what do we know about the quotient ...
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1answer
37 views

$\mathbb{Z}/2\otimes_{\mathbb{ Z}} \mathbb{ Z}$

What is the tensor product of $\mathbb{Z}/2$ and $\mathbb{ Z}$ considered as $\mathbb{ Z}$-modules? That is I want to calculate $\mathbb{Z}/2\otimes_\mathbb{ Z} \mathbb{ Z}$. Is it 0?
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12 views

Tensor product(rotation and divergence)

I want to prove this.I got struck in the last line.Can any one help me how to prove after this.
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18 views

Is tensor product commutative on orthonormal basis?

In general the tensor product $\varphi\otimes\psi$ is not commutative, but I was thinking that if I have tensor product on two orthonormal bases of Hilbert spaces are they commutative i.e is ...
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45 views

Tensor Algebra = Universal Property of FORGETFUL FUNCTOR?

Hi there in wiki the tensor algebra is defined w.r.t. the adjoint of the forgetful functor rather than the forgetful functor itself - why so? Besides, does the existence of such algebras for every ...
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21 views

Tensor product of mixed tensors

as I have understood it a mixed tensor is an element of some tensor products of the same vector space V and its dual V*. I wonder why every book wants to order them so that the V's are grouped ...
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100 views

Explicitly computing the isomorphism class of the tensor product of two finite abelian groups

How do I compute the isomorphism class of $A\otimes_\mathbb{Z} B$, where $A$ and $B$ are abelian of finite order? I can do this for a few examples, but I am unsure of how to proceed in the ...
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53 views

Tensor $\mathbb{Z}^n$ with $\mathbb{R}$

It is true that $\mathbb{R}^n\simeq\mathbb{Z}^n\otimes_{\mathbb{Z}}\mathbb{R}$? If true what would be the isomorphism?
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39 views

Cartesian & Tensor Product

What is the difference between a cartesian product and tensor product of two vector spaces $V_1$ and $V_2$ defined over same field $F$ ?
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41 views

Tensor product and Cartesian product

All the time, I encounter some literature on this it doesn't clarify a couple of things. Firstly when is Tensor product applicable and when not ? Similarly for the Cartesian product (I think more ...
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23 views

Examples where $\hat{A}$ is flat as an $A$-algebra (and $A$ is not noetherian)?

Lately I've looked a bit at $f$-adic completions of commutative rings (see for example my last 2 questions), so here's another question concerning the topic: Let $A$ a commutative ring, $f \in A$ not ...
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42 views

Characteristic Property = Universal Property?

Problem They seem to be the same -almost! But are they really or is it just unlucky accident that they look so similar however describe totally different notions? Example I was trying to set the ...
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Representation of tensor product

If we take $u\in X\otimes Y$ then we can find $\{x_{i}\}_{i=1...n}$ and $\{y_{i}\}_{i=1...n}$ linear independent such that $u=\sum_{i=1}^{n}x_{i}\otimes y_{i}.$ It is true for tensor product of three ...
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40 views

How can we show that $\operatorname{Tor}_1^A(\hat{A},G_f/G)=0$?

This question is quite closely related to my last question: Is it always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$? Let $A$ a commutative ring, $f \in A$ a non zerodivisor. Let ...
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Free vector space and its quotient

Given $\phi_1: V\times W \to F[V\times W]$ defined by $\phi_1(v,w)=(v,w)$, where $F[V\times W]$ is the free vector space. $\phi_2: F[V\times W] \to F[V\times W]/Y$ is defined by $\phi_2(v,w)=[(v,w)]$, ...
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Tensor products, existence of a unique linear map

Question: Given a bilinear map $B: V\times W\to X $, show there exists a unique linear map $T:V \otimes W\to X $ s.t. $B= T \circ \phi$ Background: We define $V \otimes W $ by F[ ...
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1answer
55 views

using the kronecker product and vec operators to write the following least squares problem in standard matrix form

I have a least squares problem with the following form: $$ \min_\mathbf{X} ~ \left\| \sum_{i=1}^n \mathbf{u}_i^\top \mathbf{X} \mathbf{v}_i - b_i \right\|^2 $$ where $\{\mathbf{u}_i\}_{i=1}^n$ and ...
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1answer
37 views

Extension and restriction of scalars and their relation to the identity functor

Let $R$ and $S$ be rings with $R \subseteq S$, and let $_S M$ be a left $S$-module. If we restrict our scalars to $R$, we naturally have a left $R$-module structure on $M$, $_R M$, given by ...
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1answer
16 views

Identification of tensor product spaces

Let $P_1, P_2$ be (probability) measures, $\Omega_1, \Omega_2 \subset \mathbb{R}^n$ . Prove that $L_{P_1 \otimes P_2}^2(\Omega_1 \times \Omega_2)$ and $L_{P_1}^2(\Omega_1) \otimes L_{P_2}^2(\Omega_2)$ ...
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1answer
46 views

Tensor Product of a Nil Algebra

Can't prove the following: Let $k$ be a field, $A$ an associative (non unital) $k$-algebra in which every element is nilpotent, and $A$ is finitely dimensional over $k$. Then for every commutative ...
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Universal property of tensor products of real Hilbert spaces

I have the following exercise where I could need some hints: Let H1, H2 be real Hilbert spaces. Prove that there is a weak Hilbert-Schmidt mapping $$ p: H_1 \times H_2 \rightarrow H_1 \otimes H_2 ...
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1answer
35 views

Homomorphisms from the base change of a module

Let $A, B$ be commutative rings with one and let $M$ be an $A$-module, $f: A \rightarrow B$ a ring homomorphism. Consider the (right) $B$-module $M \otimes_A B$. What can we say about ...
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Reduced $C^*$-algebra of a direct product of locally compact groups

Is it true that $$C^*_r(G_1\times G_2)=C^*_r(G_1)\otimes_{\min}C^*_r(G_2)$$ for locally compact groups $G_1$ and $G_2$? I have managed to prove that it holds for discrete groups (see below), but as ...
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3answers
105 views

What is $\mathbb{Z}/n\mathbb{Z}\otimes_\mathbb{Z} m\mathbb{Z}$?

I would like to know what $\mathbb{Z}/n\mathbb{Z}\otimes_\mathbb{Z} m\mathbb{Z}$ is isomorphic to, where $n,m\in\mathbb{N}$. Of course there will likely be cases depending on coprimeness and whatnot; ...
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43 views

$\frac{\mathbb{Z}}{m\mathbb{Z}}\otimes_{Z}\frac{\mathbb{Z}}{n\mathbb{Z}} \cong \frac{\mathbb{Z}}{d\mathbb{Z}}$ [duplicate]

I want to prove that $$\frac{\mathbb{Z}}{m\mathbb{Z}}\otimes_{\mathbb{Z}}\frac{\mathbb{Z}}{n\mathbb{Z}} \cong \frac{\mathbb{Z}}{d\mathbb{Z}}$$ where $m ,n \in \mathbb{N}$ and $d = \gcd(m,n)$. Any ...
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24 views

Largest? smallest cross norm? Simple question about cross norms on tensor products of Banach spaces.

This is a very simple dumb question, I'm completely new to this topic, I was reading wikipedia's entry on "Topological tensor product" and there's one thing I'm confused about. Let $A$ and $B$ be ...
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1answer
43 views

What is the difference between Cartesian and Tensor product of two vector spaces

In particular, how is it that dimension of Cartesian product is a sum of dimensions of underlying vector spaces, while Tensor product, often defined as a quotient of Cartesian product, has dimension ...
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1answer
14 views

Showing function with domain tensor product is injective? Way to do so

Show $\mathbb{Q}\otimes_\mathbb{Z}\mathbb{Q}\cong \mathbb{Q}$ as groups. I used the universal property of the canonical middle linear map to get a homomorphism ...