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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How to prove that $\phi(v\otimes f) = g(v,f)$ is injective?

Let $V$ a finite vector space and $V^{\ast}$ its dual. Let $g : V\times V^{\ast} \to \mathcal{L}(V,V)$ a bilinear map defined as follows: $$g(v,f)(w) := f(w)v.$$ To show that the map $$\phi(v\...
2
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2answers
46 views

How to show using the universal property that $V\otimes V^{\ast} \cong \mathcal{L}(V,V)$?

Let $V$ a vector space of finite dimension and $V^{\ast}$ its dual space. How to use the universal property to show that $V\otimes V^{\ast} \cong \mathcal{L}(V,V)?$ I just know that I can construct ...
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3answers
46 views

Why not use the Cartesian product for the monoidal category of modules? Why use the tensor product?

At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/...
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1answer
27 views

Unitary transform of a partially entangled 3-qubit state

Suppose we have a partially separable 3-qubit state $$φ = \left(a_0\left|0\right\rangle + a_1\left|1\right \rangle\right) \otimes \left(b_{00}\left|00\right \rangle + b_{01}\left|01\right \rangle + ...
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1answer
41 views

If $f \otimes \text{id}_{\Bbb Q}$ and $f \otimes \text{id}_{\Bbb{F}_p}$ are isomorphisms, is $f$ an isomorphism?

I would like to know the following "local-global" principle holds (all the tensors are taken over $\Bbb Z$): Let $A,B$ be two abelian groups. Assume that $f \otimes \text{id}_{\Bbb Q}$ and $f \...
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0answers
9 views

when is a separable vector a product vector?

Consider a real tensor product space $V^{(1)}\otimes V^{(2)}$, and a set of vectors of the form $a\otimes b$. A "product vector" is defined as one that separates over the tensor product, e.g. $(a+b)\...
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29 views

Tensor product and dual of vector spaces

Consider $\mathbb{F}$ the algebraic closure of a finite field with characteristic $p>0$, and let $\mathbb{F}_q$ the unique subfield of $\mathbb{F}$ with $q=p^\alpha$ elements. So if we have $J$ ...
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0answers
21 views

Nonnegative solutions of linear equations

I am wondering whether the follow proposition holds or not: Let $A$ be an $n\times k$ matrix with real entries where $k<n$. Suppose that there is some $y\geq 0$ and $y\neq 0$ with $yA^{\...
2
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2answers
35 views

Is it possible to define the tensor product of two vectors with respect to a bilinear form?

Given two vectors $\vec{v},\vec{w} \in \mathbb{R}^n$, and a bilinear form $\mathcal{B}$ represented by an $n \times n$ matrix $B$, we can define the inner product of $\vec{v}$ and $\vec{w}$ with ...
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1answer
40 views

Basis of tensor product of subspaces

Consider two vector spaces $S$ and $S\otimes S$, both of which are subspaces of $H\otimes H$, where $H$ is of $d$ dimension and so $H\otimes H$ is of $d^2$ dimension. We assume that $S$ is of $n$-...
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35 views

Finite dimensional separable algebra is étale

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every extension $L/\Bbbk$. Say it's étale if there's an extension $L/\Bbbk$ such that $L\otimes_\Bbbk A\cong \prod_1^nL$. Here'...
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1answer
194 views

Why does this ring have rank $k!$?

Let $R$ be any ring free of rank $k$ over $\mathbb{Z}$ having non-zero discriminant. Let $R^{\otimes k} = R \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} R$. Then $R^{\otimes k}$ is a ring of rank ...
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1answer
25 views

A basis for a tensor product space where the tensor elements are linearly dependent

Say I have a space $V^{(1)}$ with basis $\{a_i \}$ and $V^{(2)}$ (with dimensions $d_1$, $d_2$ respectively) with basis $\{b_j\}$. Clearly the vectors $\{a_i\otimes b_j\}$ are a basis for $V^{(1)}\...
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16 views

Proof of tensor product of two vector spaces

We know that tensor product of V and W(assume both are vector space over F)can be identified with the linear functionals from V cross W to F.But when we actually try to construct explicitly the tensor ...
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51 views

A nonnegative vector orthogonal to tensor space

Let $S$ be a subspace over $\mathbb{R}$. For a vector $v$, $v\geq 0$ means each entry of $v$ is nonnegative. Does it hold that, for any $v\geq 0$ such that $ v\bot S^{\otimes n} $, there is some $v'\...
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10 views

Tensor product of graded maps

Let $M$ and $N$ be graded R-modules. Then $M= \oplus_{k \in \mathbb{Z}}M_k$ and $N= \oplus_{k \in \mathbb{Z}}N_k$. Then $M \otimes_R N$ is also a graded R module whose kth component is $(M \otimes_R ...
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1answer
25 views

Generic Element of Compositum of Two Fields [duplicate]

I'm interested in understanding compositum of general fields better. Assume we have $\Omega/K/F$ and $\Omega/L/F$ field extensions, and consider the composite $KL$. It seems to me that every $a \in ...
2
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0answers
26 views

addition of tensors

The tensor algebra over a $n$-dimensional vector space $E$ is defined as the direct sum: $$\bigotimes E:=\bigoplus_{r,s>0}\Big(\bigotimes\nolimits_r^sE\Big)$$ An element of $\bigotimes E$ is ...
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1answer
25 views

Tensor products $L_1(\mu)\widetilde{\otimes}_{\pi} L_1(\nu)$ and $L_1(\mu)\widetilde{\otimes}_{\varepsilon} L_1(\nu)$

Can anybody enlighten me, where the tensor products of the spaces of summable functions $L_1(\mu)\widetilde{\otimes}_{\pi} L_1(\nu)$ and $L_1(\mu)\widetilde{\otimes}_{\varepsilon} L_1(\nu)$ are ...
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1answer
48 views

Prove that $A \otimes_R B \cong (A \otimes_{\mathbb{Z}} B)/ H$

I am working on the following problem: Let $A \in Mod-R$ and $B \in R-Mod$. Prove that $A \otimes_R B \cong (A \otimes_{\mathbb{Z}} B)/ H$ where $H=\langle ar\otimes_{\mathbb{Z}}b - a\otimes_{\mathbb{...
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60 views

Ideals and Tensor Products

I'm reading Osbourne's Basic Homological Algebra, and on page 18 he has this situation where we've got a ring $R$ and a right-ideal $I$, and some left $R$-module $B$. He says $I\otimes B$ is not a ...
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1answer
26 views

Is the tensor algebra functor a strong monoidal functor?

Let $K$ be a commutative ring. Is it true that $T(V \oplus W) \cong T(V) \otimes_K T(W)$ as $K$-algebras for any $K$-modules $V$ and $W$? The reason I ask is that I have heard it mentioned (I don't ...
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1answer
24 views

Question about $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$.

I am confused about the set $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$... Could someone please explain me why this corresponds to the set of pairs of symmetric $3$ by $3$ matrices? Thank you!
2
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1answer
63 views

What is $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p(X)} \mathbb{F}_p(\sqrt{X})$?

I am trying to understand what $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p (X)} \mathbb{F}_p(\sqrt{X})$ is. $\mathbb{F}_p(\sqrt{X})$ is the field of rational functions in $\sqrt{X}$. What is it ...
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2answers
499 views

How to understand “tensor” in commutative algebra?

Tensor is sure an important concept in commutative algebra, but the definition is kind of abstract, so is there any way to understand it which is easier? Thanks advance! The definition I see is the ...
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1answer
19 views

Norm of the gradient of a vector field in Cartesian versus Cylindrical coordinates

It is well known that for a vector $\textbf{v}=R\textbf{e}_r+\Theta\textbf{e}_{\theta}+Z\textbf{e}_z$, its 2-norm is $\|\textbf{v}\|_2=\sqrt{R^2+Z^2}$ instead of $\sqrt{R^2+\Theta^2+Z^2}$. Now, for a ...
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25 views

Show that $\mathbb{C}\otimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\times\mathbb{C}$ [duplicate]

Show that $\mathbb{Q}(\sqrt{2})\otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{3})\simeq\mathbb{Q}(\sqrt{2},\sqrt{3})$ and $\mathbb{C}\otimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\times\mathbb{C}$ I approached ...
2
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0answers
26 views

Finite Dimensional Representation of Lie Algebra.

Let $V, W, U$ be finite dimensional representations of a lie algebra $\mathfrak{g}$. Show that $\hom(V \otimes W, U) \cong \hom (V, U \otimes W^*)$. I think I have to use the enveloping algebra of ...
1
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1answer
43 views

Adjointness of Hom and Tensor for non commutative ring

Let $M$ be an $(A,B)$-bimodule, let $N$ be a $(B,C)$-bimodule, and let $K$ be an $(A,C)$-bimodule. Then $$\operatorname{Hom}_C(M \otimes_{B}N,K) \cong \operatorname{Hom}_B(M,\operatorname{Hom}_C(N,K))$...
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1answer
30 views

conversion between cylindrical local basis (coordinates) and Cartesian local basis (coordinates)

We know that the basis vectors $\{\textbf{e}_r,\textbf{e}_{\theta},\textbf{e}_z\}$ for cylindrical coordinates and the basis vectors $\{\textbf{e}_x,\textbf{e}_y,\textbf{e}_z\}$ for Catesian ...
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0answers
36 views

Tensor product for vectors

This might be a simple question but I'm struggling to understand the tensor product of two vectors. From what I understand if $\vec{v}$ and $\vec{w}$ have dimensions $2$ and $3$ respectively. Then the ...
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0answers
31 views

Applications of tensor product of graphs (modelling of Internet Graphs)

I was going through the book Handbook of Product Graphs, by Richard Hammack, Wilfried Imrich, Sandi Klavžar. Somewhere in book, they mentioned the following lines : One of the applications of tensor ...
1
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1answer
45 views

A more useful basis under extension of scalars

Consider a field extension $L/K$ and a $K$-vector space $V$. The tensor product $L \otimes_K V$ with $\lambda (\alpha \otimes v) := (\lambda \alpha) \otimes v$ where $\lambda,\mu \in L$ makes it an $...
2
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1answer
35 views

Exterior algebra of a ring

In the book "Cohen-Macaulay rings" by Bruns and Herzog, the quick introduction of tensor algebra and exterior algebra left me a bit bewildered. After referring to the section on tensor algebra ...
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0answers
31 views

Balanced Tensor Product of Module Categories

Let $C$ be a $k$-linear ($Vect_k$-enriched) monoidal category and consider the 2-category $Mod_{C}$ of $k$-linear $(C,C)$-bimodule categories in the sense of Ostrik (https://arxiv.org/abs/math/0111139....
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1answer
30 views

Let $V$ and $W$ be vector spaces over $\mathbb{C}$. Show that $0\otimes w = v\otimes 0 = 0 \in V \otimes W$.

Algebraically, the vector space $V \otimes W$ is spanned by elements of the form $v \otimes w$, and the following rules are satisfied, for any scalar $c$. The definition is the same no matter which ...
3
votes
1answer
72 views

Dual space of polynomial algebra

Let $k$ be an infinite field and let's consider the ring $R=k[x_1,\dots,x_n]$. This ring has a structure of $k$-vector space (or a $k$-algebra). I am interested to know about the structure of the ...
4
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3answers
60 views

Tensor product $Z[1/2]\otimes Z/3$

Compute $\mathbb{Z}[1/2]\otimes_{\mathbb{Z}} \mathbb{Z}/3$ My initial instinct was that it is equal to $\mathbb{Z}/3[1/2]$, but I couldn't show this is correct by the universal propert. Hence, I ...
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2answers
53 views

how the Kronecker product is a tensor product?

I have seen this question but still don't understand how the Kronecker product is a tensor product. If $T_1 : V_1 \to W_1$ and $T_2 :V_2 \to W_2$ are linear transformations their matrix ...
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0answers
27 views

linear decompositions of tensor product vectors

I have a vector space $V^{(1)} \otimes V^{(2)}$, a set of vectors $\{s_i\} \in V^{(1)}$ and $\{s_j\} \in V^{(2)}$, that span the respective spaces such that any one vector, e.g. $s_{i'}$, can be ...
2
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1answer
52 views

Can we write $\mathbb{Q}[X]/(X^2 - 2) \otimes_\mathbb{Q} \mathbb{Q}[X]/(X^2 + 1)$ as a product of fields?

Sometimes we can write an algebra as products of fields using Chinese Remainder Theorem. For example, $\mathbb{Z}_6 \simeq \mathbb{Z}_2 \times\mathbb{Z}_3$. Another example would be $\mathbb{F}_2 [X]/(...
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3answers
62 views

Show that $\mathbb{Z}/a \otimes \mathbb{Z}/ab$ is not zero.

This question is motivated by the fact that, if $\gcd(a,b) = 1$ then $\mathbb{Z}/a \otimes \mathbb{Z}/b$ is zero tensor product. I would like to show that $\mathbb{Z}/a \otimes \mathbb{Z}/ab$ is ...
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0answers
12 views

Does tensor product with $L_p$ operator algebra preserve exact sequences?

By $L_p$ operator algebra I mean a closed subalgebra of the algebra of bounded linear operators on some $L_p$ space where $p\in(1,\infty)$. There is a notion of tensor product of $L_p$ spaces (as ...
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0answers
35 views

Application of tensor product of graphs in real life.

I was going through the book HANDBOOK OF PRODUCT GRAPHS by Richard Hammack, Wilfried Imrich, and Sandi Klavzar. In the preface section, application of direct product of graphs is mentioned. I am ...
2
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0answers
30 views

Orientation of vector space given that of its subspace and the associated quotient

In D. Freed's lecture notes pp.2 he mentions the following way to define the orientation of a vector space given that of its subspace and the associated quotient. let $0 \to V' \to V \to V'' \to 0$ ...
6
votes
3answers
105 views

Understanding the tensor-hom adjunction intuitively

I'm currently trying to teach myself some category theory. Recently, I learned that the tensor product is left adjoint to the hom functor in suitable categories, e.g. vector spaces with linear maps, i....
5
votes
1answer
49 views

Difference between $\mathbb{Q}[X]/(X-1) \otimes_\mathbb{Q} \mathbb{Q}[X]/(X+1)$ and $\mathbb{Q}[X]/(X-1)\otimes_{\mathbb{Q}[X]}\mathbb{Q}[X]/(X+1)$?

The original problem actually wants me to find which one is a zero module. But first, what is the difference between $\mathbb{Q}[X]/(X-1) \otimes_\mathbb{Q} \mathbb{Q}[X]/(X+1)$ and $\mathbb{Q}[X]/(X-...
0
votes
0answers
11 views

Hausdorffness of projective tensor product of locally convex space

I do not see why the following is true: Given two locally convex t.v.s. E and F, if the projective tensor product is Hausdorff then both E and F are Hausdorff. The converse holds and the proof is ...
1
vote
2answers
63 views

Show that $M\otimes N$ is isomorphic to $N\otimes M$

I want to prove the following: Let $A$ be a ring and $M,N$ be $A$-modules. Show that the tensor products $M\otimes N$ and $N\otimes M$ are isomorphic. I have consulted this question but it did ...
4
votes
1answer
49 views

How to represent matrix multiplication in tensor algebra?

How can we represent matrix multiplication in tensor algebra? Even if we assume all matrices represent contravariant tensors only, clearly matrix multiplication does not correspond to the ...