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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13 views

Is $p\in B(\mathbb{C}^4)$ a s.o.t-limit of a sequence $(a_n\otimes b_n)_{n\in\mathbb{N}}\subseteq B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)$?

Let $L(H)$ the bounded linear operators on a hilbert space $H$. I proved that the inclusion $$i:B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)\hookrightarrow B(\mathbb{C}^4)$$ is not surjective: take $p=\...
0
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1answer
30 views

Is this tensor contraction correct?

If I start off with the $\left(p,q\right)$-tensor given by $$T_{i_1,\dots,i_p}^{j_1,\dots,j_q}e_{i_1}\otimes\dots\otimes e_{i_p}\otimes\varepsilon^{j_1}\otimes\dots\otimes\varepsilon^{j_q}$$ and I ...
1
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0answers
34 views

Understanding isomorphism of Hom's

During some reading I came across the statement $$Hom_R(M,N) \otimes R/I \cong Hom_{R/I}(M/IM,N/IN) $$ where $M,N$ are $R$-modules and $I$ an ideal of the commutative ring $R$. This is proved, under ...
3
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2answers
101 views

Geometrical Interpretation of Tensors (intuition)

How would one describe to a non-mathematician (an undergraduate physicist actually- so please do use mathematics-i am not even sure if they can be described without mathematics!) what do tensors ...
2
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1answer
20 views

Does this symmetric rank-3 tensor vanish?

Suppose we have a rank-3 tensor $T$ on some vector space $\mathbb{V}$. We can view $T$ as a map: $$T: \mathbb{V} \times \mathbb{V} \times \mathbb{V} \to \mathbb{R},$$ which maps triples of vectors in ...
1
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1answer
54 views

Tensor product of coordinate rings corresponds to pullback

Here in Milne's notes on algebraic geometry, he proves that if $k$ is an algebraically closed field, and $A$ and $B$ are reduced finitely generated $k$ algebras, then $A \otimes_k B$ is reduced. (This ...
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23 views

Tensor products in separable Hilbert spaces

In a variety of scientific publications, I have come across the use of tensor products of random Hilbert-Schmidt operators defined on separable Hilbert spaces. Let us introduce some notations. Define ...
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23 views

Commutation of Tensor Products as operators

Suppose I have unitary operators $$A: \mathbb{C}^{2^k} \rightarrow \mathbb{C}^{2^k}$$ $$B: \mathbb{C}^{2^j} \rightarrow \mathbb{C}^{2^j}$$ For some $k,j \in \mathbb{Z}, j,k \ge 0$. How do show that ...
0
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26 views

$\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert space

I want to describe $\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert spaces; $\mathbb{C}^n\otimes \mathbb{C}^m$ is endowed with the scalar product $\langle x\otimes y, x'\otimes y'\...
0
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28 views

Tensor Product of Algebras Commutes with Products?

It seems to me that if $R$ is commutative and $A,B,C$ are (associative, unital) $R$ algebras then $(A\times B)\otimes_R C\approx A\otimes_RC\times B\otimes_RC$. On the level of $R$ modules, we can use ...
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14 views

How to write clebsch gordan decomposition using tensor notation

Let be $G$ a Lie Group and $\textbf{N}$ its complex representation. It is well know that any state $|\ ab\ \rangle\in \textbf{N}\otimes\textbf{N} = \oplus_I\textbf{r}_I$ can be decomposed through the ...
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1answer
54 views

Linear algebra prerequisites for abstract algebraic geometry

I'm interested in what linear/multilinear algebra does one need to study algebraic geometry(following EGA and Harthshorne). Texts I have in mind are like "Foundations of algebraic geometry" by Ravi ...
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2answers
53 views

eigenvalues of sum of tensor product of matrices

Lets say $A, B$ are rank 1 hermitian real matrices. It follows immediately that $A\otimes A$ and $B\otimes B$ individually are rank 1. My question is how does the largest eigenvalue of $A+B$ relate ...
0
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1answer
43 views

Prove the pullback of the wedge product is the wedge product of the pullbacks.

Let $F:V \rightarrow W$ be a linear map. Show that $F^{\ast}(\omega \wedge \eta)=(F^{\ast}\omega) \wedge (F^{\ast}\eta)$ for all $\omega \in \Lambda^{p}(W) , \eta \in \Lambda^{q}(W)$. Where $F^{\ast}...
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18 views

Correspondence between tensors and multivectors

In one of its definitions the Clifford algebra is defined as a quotient space of a (infinite dimensional) Tensor algebra. The question is: Given the metric signature and the tensor $T$ and its ...
3
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1answer
42 views

Question about two ways to induce an inner product on $S^2V$

$\newcommand{\til}{\tilde}$ Let $(V,g)$ be an $n$-dimensional inner product space, and let $S^2V^*$ be the symmetric algebra. I am familiar with a natural way to endow $S^2V^*$ with an inner product ...
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25 views

Equivalent conditions for a short pure exact sequence

Let $L$, $N$, and $M$ be right $R$-modules and let $\widehat{L}=\mathrm{Hom}_{\mathbb{Z}}\left(L,\mathbb{Q}/\mathbb{Z} \right)$ ($\widehat{N}$ and $\widehat{M}$ are defined analogously). Show that the ...
1
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1answer
23 views

Definition of Outer Product of (abstract) vectors

I was reading an article from the American Mathematical Monthly on the Caratheodory derivative for functions of several variables, and in one of the proofs the authors construct a linear ...
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0answers
12 views

Ordering of basis elements of a Lie-group representations tensor product

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$ \textbf{N}\otimes\bar{\textbf{N}} = \bigoplus_J \textbf{r}_J $$ where $\textbf{r}_J$ ...
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40 views

Hadamard product involving operators

If we have two matrices $A=(a_{i,j})_{i,j}$, $B=(b_{i,j})_{i,j}$ representing linear and continuous operators from $\ell^2$ to $\ell^2$, it is known that the Hadamard product of them, $A\ast B=(a_{i,j}...
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35 views

Can this matrix equation be solved?

I have a matrix equation of the form [Z][C][Z] - [Z][D] = [A][Z] - [B] and I need to solve for [Z]. Is it possible? If so, what is the solution? The matrices are all general, complex, with no ...
2
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1answer
36 views

Covariant Contravariant approach for Tensors

I'm reading a book on Geometry from the '70s and when speaking about Tensors it defines them starting from the covariant and contravariant commutation rule. I know this definition was quite widespread ...
0
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2answers
39 views

Tensor algebra with semidirect product

I am having trouble parsing the following problem: Let $F$ be a field, $A$ an $F$-algebra with 1 (not necessarily commutative) and let $M$ be an $F$-vector space which is also an $A$-bimodule. Let $A\...
4
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1answer
68 views

Differing definitions of a connection on a vector bundle

My lecture notes define a connection on a vector bundle $\pi:E\rightarrow{M}$ to be an $\mathbb{R}$-linear map: \begin{equation} \nabla:\Gamma(E)\rightarrow\Gamma(T^*M\otimes{E}) \end{equation} ...
2
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1answer
32 views

Construction of Tensor Product on $\mathbb R^2$

I am trying to construct a tensor product on $ \mathbb R^2$. I have defined a bilinear map by $\phi:\mathbb R \times \mathbb R\to \mathbb R $, as $\phi(x,y) = xy$ Which clearly is a bilinear map. ...
3
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60 views

Understanding the construction of Exterior Algebra

Background The tensor space of type $(r,s)$ associated with $V$ is the vector space $$\underbrace{V\otimes \ldots \otimes V}_{\text{r copies}} \otimes \underbrace{V^* \otimes \ldots \otimes V^*}_{\...
2
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2answers
55 views

Using Universal Mapping Property of Tensors to show that $\dim V \otimes W=(\dim V)(\dim W)$

Background Let $V,W$ be finite dimensional vector spaces, $V^*$ the dual space of $V$ and $\mathrm{Hom} \ (V,W)$ the vector space of all linear transformations from $V$ to $W$. The universal mapping ...
2
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2answers
43 views

Why does this tensor product has exactly 2 elements?

I am trying to prove that $(\mathbb{Z}/10\mathbb{Z})\otimes (\mathbb{Z}/12\mathbb{Z}) \cong (\mathbb{Z}/2\mathbb{Z})$. I understand now that every element in $(\mathbb{Z}/10\mathbb{Z})\otimes (\...
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1answer
26 views

What are the submodules of $M⊗K$?

Let $k$ be a field and $K$ is a extension of this field. And let $A$ be a finite type $k$-algebra, and $M$ be a finitely generated module over $A$. Then, is the form of submodules of $M⊗_kK$ always $...
2
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0answers
45 views

Help with a statement in P.286 of Hatcher

In P.286 of Hatcher's 'Algebraic Topology', it is stated that $\alpha_{p^i}$ is primitive in $ \bigotimes_{i\geq 0} \mathbb{Z}_p[\alpha_{p^i}]/(\alpha_{p^i}^p)$, where 'primitive' means the coproduct $...
1
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1answer
54 views

Is there a relation between Cartesian and tensor product of function spaces and function factorizability

H1 and H2 are two Hilbert spaces represented by a function space, say f1(x1) and f2(x2) are its vectors. If H3 is tensor product of H1 and H2 I assume one can say that f(x1,x2) now represents vectors ...
1
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1answer
46 views

When (and why) tensor product? When dot product?

This looks a lot like physics, but it is actually a math question! I will be omitting unnecessary constants for simplicity so the units might be off. I want to reduce the equation $-i\omega \vec{j}(...
8
votes
2answers
90 views

Is the tensor product over $B$ of two flat $A$-modules flat over $A$?

Given a morphism of commutative rings $A\to B$ such that $B$ is a flat $A$-module and given $M$, $N$ two $B$-modules flat as $A$-modules, is the tensor product $M\otimes_B N$ flat over $A$?? The ...
0
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0answers
15 views

Tensor product on a hilbert space

If I have the expression: $\langle\phi|\hat{A}$ $ \otimes \hat{I}|\phi\rangle$ $ $ $ $$ $(*) where $\hat{A}$ is a linear operator $\hat{I}$ is the identity operator and $| \phi \rangle \...
0
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1answer
26 views

Tensor prodct on Hilbert Space

How do I show that $\langle \phi|I \otimes I|\phi\rangle=1$ ? where: $I$ is the identity operator and $\phi \in \mathbb{C^2}\otimes\mathbb{C}^2$
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13 views

Tensor product and conjugate space

Let $H$ and $K$ be Hilbert spaces and $\overline{K}$ be the conjugate space of $K$, i.e., the elements of $\overline{K}$ are the same of $K$, the sum and norm are the same, and the product rule is ...
0
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1answer
49 views

Tensor product of operators

How do I show that: $(\hat{\sigma} $ $\otimes $ $\langle \Phi|\hat{I} )(\hat{\sigma} $ $\otimes $ $ \hat{I})|\Phi\rangle$ $ =1 $ (The parenthesis many not be strictly correct here) ...
0
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2answers
85 views

Assign a unique number to every linear form $\varphi : \mathbb R^n \to \mathbb R$, has this number a geometric interpretation?

If $V$ is a finite-dimensional vector space of dimension $n$, denote the space of all alternating $k$-fold multilinear maps (also called alternating $k$-tensors) by $\Lambda^k(V)$. Then $\dim \Lambda^...
4
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2answers
77 views

Is the tensor product of non-commutative algebras a colimit?

For $R$ a commutative ring, the tensor product of $R$-algebras is the coproduct in the category of commutative $R$-algebras. In the noncommutative case it is no longer the coproduct in the category of ...
1
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1answer
41 views

Geometric generic fibre

I have a question concerning the following exercise in Hartshorne: The inclusion of $k[s]$ into $k[s,t]/(s-t^2)$ induces a morphism of the corresponding affine schemes $X\to Y$. The exercise itself ...
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23 views

Does every tensor norm satisfy one half of the crossnorm property

Suppose $(H,\|\cdot\|_{H})$ and $(G,\|\cdot\|_{G})$ are normed vector spaces. Suppose $\|\cdot\|$ is a norm on their algebraic tensor space $H\otimes G$. Do we have $$ \|h\otimes g\|\leq C\|h\|_{H}\|g\...
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0answers
44 views

If $E$ is a finitely presented left $A$-module, then tensor product by E commutes with direct product.

Please can anyone help me to prove this statement: If $E$ is a finitely presented left $A$-module, then for every family $(F_i)$ of right $A$-modules the canonical homomorphism $\phi: E\...
5
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1answer
24 views

Associating a $m-1$-tensor on $\mathbb R^m$ to an element of $\mathbb R^m$.

Show that for every alternating $m-1$-tensor on $\mathbb R^m$ there exists a unique $v \in \mathbb R^m$ such that for every linear function from $\mathbb R^m$ into $\mathbb R$ and for every $v_1, ...
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2answers
29 views

Verify a Tensor Differential Identity

I would like to show that the ID below holds: $$(n \cdot \nabla n)=1/2 \nabla(n \cdot n)-(n \times(\nabla \times n))$$ Using the Einstein notation I've come up with: $$(n \times(\nabla \times n))_i=...
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0answers
14 views

Matrices over tensor product of Banach algebras

Suppose that $A$ is a Banach algebra, or even a $C^*$-algebra, and $B$ is a closed subalgebra of $B(L^p)$ for some $L^p$ space. In particular, $M_n(B)$ has a canonical matrix norm. Is there some ...
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1answer
60 views

Tensor product group representations and spaces of intertwiners.

Let $V_{1}$, $V_{2}$, $W_{1}$, and $W_{2}$ be the carrier spaces of representations of some finite group $G$. Suppose also that $G$ acts trivially on $V_{1}$ and $V_{2}$. I would like to prove the ...
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2answers
37 views

Atiyah and Macdonald's proof of the existence of the tensor product

I have a question regarding the proof of the proposition 2.12 in Atiyah and Macdonald's book. They say: " Let C denote the free A-module $A^{(M \times N)}$. The elements of C are formal linear ...
4
votes
1answer
92 views

In which sense is composition a tensor product

Let $\Phi\colon U\to V$ and $\Psi\colon V \to W$ be linear operators, and consider their composition $$ \Psi\circ \Phi $$ The operation, $$\circ:\mathcal{L}(U,V)\times\mathcal{L}(V,W)\to \mathcal{L}(...
1
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1answer
49 views

Is $M_R\otimes _R {_R}N\cong M_{\mathbb Z}\otimes_{\mathbb Z} {_{\mathbb Z}}N$?

Suppose $M$ is a right $R-$module and $N$ is a left $R-$ module. Also $M$ and $N$ are naturally $Z-$ module, both in left and right side. So we will denote $M_R$, $M_{\mathbb Z}$, and $_RN, _{\mathbb ...
1
vote
1answer
55 views

Tensor product of purely inseparable extension

I would like to understand the tensor product of $A=\Bbb F_2(\sqrt{t})\otimes_{\Bbb F_2(t)}\Bbb F_2(\sqrt{t})$. The extension $L/k:=\Bbb F_2(\sqrt{t})/\Bbb F_2(t)$ is a finite extension of degree $2$...