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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Tensor algebra becomes a graded $R$-algebra short proof

I had a post proving that the tensor algebra becomes a graded ring, i have come up with a simple approach that goes as follows: Proposition: The tensor algebra $T(M)$ with multiplication defined ...
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34 views

Tensor algebra becomes an R-algebra. Theorem, Proof, Dummit and Foote

I have the definition of tensor algebra as follows: $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$, where $M$ is an $R$ module, where $R$ is commutative and contains the element $1$. Finally $T^k(M) = ...
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46 views

Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$. Consider the tensor product of these maps $L_1\otimes ...
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27 views

Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much ...
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8 views

Tensor products of Lipschitz functions

I have encountered a problem on which I am sure there is some background, which unfortunately I don't know anything about (so that I don't even know where to start). Let $(M, d_M)$, $(N, d_N)$ be ...
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11 views

Injective homomorphism on tensor product

I am currently attempting the following: Find (cyclic) $\mathbb{Z}$-modules $M, N, P$ and an injective homomorphism $f: M \rightarrow N$ s.t. $g: M \otimes_{\mathbb{Z}} P \rightarrow N ...
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Explicit example of tensor norms

I can't find any example anywhere on the web where someone actually evaluates a non-trivial tensor norm. So I'm wondering about the simplest non-trivial case. Let $X$ be $\mathbb R^2$ with the ...
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43 views

Is there an “internal” definition of the tensor product?

We have the following "internal" definition of the direct sum: A vector space $V$ with subspaces $S,T$ is said to be the direct sum of $S$ and $T$ if $S + T = V$ and $S \cap T = \{0\}$. (Of course ...
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44 views

Prove that the following is a non zero tensor.

I'm asked to prove that the ideal $I=(x,y)$ in $R=k[x,y]$ is not a flat R-module. My approach was to use the exact sequence $$0\rightarrow I \to R \to R/I \to 0$$ to induce a non injective map ...
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10 views

When do injective and projective tensor norms agree?

For $C^*$-algebra tensor products, one talks about the min and max tensor norms, and they agree when one of the $C^*$-algebras is nuclear. For general Banach algebras, what is the analog of ...
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20 views

Module homomorphism on the algebraic closure of $\mathbb{Z}_p$

Let $k$ be the algebraic closure of $\mathbb{Z}_p$ and $k^*=k-\{0\}$ be a multiplicative group. For a finite abelian group (i.e. a finite $\mathbb{Z}$-module) $G$, compute the following: ...
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derivative of a tensor function

If e is a second order tensor and it's symmetrical, assuming abs() is a tensor function such that returns all the components of e be positive, I am interested the derivatives of the function abs(e) ...
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19 views

Pullbacks of symmetric tensors commute with products

The problem: Show that $$F*(AB)=(F*A)(F*B)$$ where F is a smooth map from a smooth manifold M to another smooth manifold N, A and B are symmetric tensor fields on N, and $F*$ denotes the pullback ...
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75 views

How are these definitions of the inertia tensor the same?

I'm looking for some help in understanding the inertia tensor (not the physics, just the math). I'm trying to figure out how to convert between the wedge product and tensor product definitions. ...
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17 views

Proving existence of Hermitian Adjoint in unusual way

For a map $T:V\rightarrow V$, we define the Hermitian adjoint to be the unique $T^*:V\rightarrow V$ such that $\langle Tu,v\rangle = \langle u, T^*v\rangle$. There are two things I'm required to ...
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1answer
38 views

How do I show this relation between exterior product and the projection of a tensor product

I have troubles understanding this whole problem starting at the definition. We have defined the exterior product as follows: If $\alpha = \pi (a) \in \bigwedge^pV$ and $\beta = \pi(b) \in ...
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1answer
18 views

injectivity. identity map of a ring to its tensor product

Let $B$ be an $A$-algebra, $f: B \rightarrow B \otimes_{A} B$ is defined by $f(b)=b\otimes 1$. Is $f$ injective? I know the definition of tensor product and started from representing as $(b,1)=\sum ...
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47 views

Length of tensor product of finite length modules is finite

Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M\otimes_R N$ has finite length, and $l(M\otimes_R N) \le l(M)l(N)$. I know the question has been posted ...
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14 views

Isomorphism of tensor product involving a principal ideal

This question arose when dealing with a long exact sequence of Tor. Let $R$ be a (not necessarily commutative) ring, $g$ a central element of $R$ and $M$ a right $R$-module. We have an exact sequence ...
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52 views

Module structure of base extension via tensor product

Let $A,B$ be commutative rings. Defining a product of $B\otimes_{A}B$ as $(b_1 \otimes b_2)\cdot (b_3 \otimes b_4)=(b_1b_3)\otimes(b_2b_4)$, this becomes a commutative ring. Defining $b\cdot(b_1 ...
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95 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley ...
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17 views

rank 3 tensor product

A can be any elements you choose. I suppose the identity matrix would look like this and the augmented 3-hypermatrix would look like A below.
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21 views

Intrinsic Definition of $^\sigma\alpha$, where $\alpha$ a covariant $k$-tensor and $\sigma\in S_k$

Let $V$ be a finite dimensional vector space and $\mathcal T^k(V^*)$ be the set of all the covariant $k$-tensors on $V$. The symmetric group $S_k$ acts on $\mathcal T^k(V^*)$ as follows: Given ...
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33 views

What is the prerequisiste to study Tensors for application in signal processing?

I want to study Blind Source separation in signal processing for this I need to study Tensors and have a basic idea about rank, border rank and other concepts. Right now I am studying from ...
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Complex euclidean tensor products

Say you have Euclidean vectors $\mathbf{a}=a_i \mathbf{p}_i$ and $\mathbf{b}=b_j \mathbf{q}_j$ in $\mathbb{R}^3$, with bases $\mathbf{p}_i$ and $\mathbf{q}_j$. Then you could use a typical inner ...
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Intuition about $v\otimes w$

In Physics and Differential Geometry usually tensors of type $(k,l)$ on a vector space $V$ over $\mathbb{F}$ are defined as multilinear functions $$f : \underbrace{V\times\cdots\times V}_{k \ ...
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1answer
14 views

Divergence of a vector tensor product / outer product:

I'm currently studying the derivation of the RANS (Reynolds Averaged Navier Stokes) equations, used in the study of turbulence, and I've stumbled upon a step wich I don't understand very well. The ...
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1answer
32 views

Computing bases for direct, wedge, tensor products, etc., of given vector spaces

I am filled with all kinds of vector space and I want to make sure I understand the basis for each kind of vector space. Suppose $\{v_i\}_{i=1}^n$ is the basis for vector space $V$, $\{w_j\}_{j=1}^m$ ...
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1answer
42 views

Why $M \otimes M$ does not have a ring structure?

I am reading some section about tensor algebras, and I don't have clear the idea on why $M \otimes M$ dont have a ring structure, where $M$ is an $R$-module. R is commutative and $1 \in R$. So far my ...
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14 views

use universal properties to prove the existence of isomorphism

Use universal properties to prove that for a finite dimesional vector space $V$ and $W$ there is a canonical isomorphism: $$\bigwedge^2(V\oplus W)\to \bigwedge^2V\oplus(V\otimes W)\oplus\bigwedge^2W$$ ...
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28 views

Reference request: extending tensor product of modules

I'm looking for a reference to a construction similar to the following. I have a right R-module, $A_{K'}$, and a left R-module, $_KB$, where $K$ and $K'$ are fields and $K'\subset K$. I want to take ...
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1answer
39 views

How to prove tensor product is exact when acted on split short exact sequence?

I know tensor product is right exact, but I can't figure out why it's exact when it is acted on a split short exact sequence. In addition, can you give an example that tensor product acts on a short ...
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1answer
45 views

Kernel of a map on tensor product of modules

Let $M,N, P$ be $A$-modules, and let $f:M \otimes N \to P$ be an $A$-homomorphism. If $m \otimes n \in \ker f$ implies $m\otimes n =0$ for all $m\in M, n\in N$, does it follow that $\ker f=0?$ For ...
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22 views

use universality property to prove the existence of isomorphism

Suppose $V$ and $W$ are finite-dimensional vector spaces over a field $K$. Use the universality property of tensor products to show that there exists a canonical isomorphism $f:V^{*}\otimes W\to ...
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66 views

Adjoint functor to an R-algebra only “remembering” itself as a ring

I have been wondering this question while trying to comprehend adjoint functors and the various definitions. If you let $$F:\mathbf {R\text - Alg}\to \mathbf {Ring}$$ be the functor that sends ...
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32 views

Why the following is not a tensor?

Say we have an arbitrary coordinate system, in which a position vector is represented by: $$\vec V=Z^i\vec Z_i$$ Where $\vec Z_i$ is a covariant basis. Now $\partial {\vec Z_i}/\partial Z^j$ term has ...
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1answer
44 views

Prove that $Ker(g \otimes k)= Im(f \otimes 1_{N}) + Im (1_{M} \otimes h)$

Suppose we have two short exact sequences: $$0 \to M' \mathrel{\overset{f}{\to}} M \mathrel{\overset{g}{\to}} M'' \to 0 $$ in Mod-R $$0 \to N' \mathrel{\overset{h}{\to}} N \mathrel{\overset{k}{\to}} ...
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18 views

Prove that $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup of $A$

Let $A$ be a finite abelian group of order $n$ and let $p^k$ be the largest power of the prime $p$ dividing $n$. Then $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup ...
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2answers
32 views

Action of universal R-matrix of U_q(sl_2)

My question is really simple but requires a few definitions. No special knowledge of quantum groups should be needed, it is more about tensor algebra. Let $q \in \mathbb{C}$ with $q \neq 0, \pm 1$. ...
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1answer
21 views

The rank of a tensor is not invariant under isomorphism

Let $V,W$ be two $K$-vector spaces; for $a\in V\otimes W$ define \begin{equation*} rk(a)=\min \left\{r|a=\sum_{i=1}^r\alpha_iv_i\otimes w_i\text{ for some }\alpha_i\in K,v_i\in V,w_i\in W\right\}. ...
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40 views

Partial Trace of Density Operator

Before stating my question I present my motivation: to learn more about the tensor product. Now, quantum mechanics assigns a Hilbert space to each physical system as a postulate of the theory. ...
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1answer
32 views

Natural Ordering of the Class of Hermitian Preserving Maps

I am trying to understand Man-Duen Choi's remark 3 in his paper Completely Positive Linear Maps on Complex Matrices: For a linear map $\Phi : \mathcal{M}_{n} \to \mathcal{M}_{m}$, it is obvious that ...
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20 views

Tensor product of algebras and generating sets.

Let $A$ be a module over $k$ generated by $x$ and $y$. The generating set for $A \otimes_k A$ is $\{x \otimes x, x \otimes y, y \otimes x, y \otimes y\}$. But does this still hold if $A$ is an algebra ...
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37 views

What is this notation $\odot$ for?

(Note that symmetric algebra and symmetric tensor do not coincide when the characteristic is not $0$.) I'm reading this aricle:http://en.m.wikipedia.org/wiki/Symmetric_tensor And here it defines ...
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1answer
23 views

Testing for decomposable tensors

Let $V$ be a vector space of dimension 3 with the basis {$e_1,e_2,e_3$} and let $W$ be a vector space of dimension 2 with basis {$f_1,f_2$}. Which of the following tensors are decomposable (i.e. of ...
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1answer
11 views

Is it okay to define k-th symmetric power of $M$ in this way?

I want to define the tensor algebra and related algebras in a very formal way. I will illustrate how I tried to define algebras below. Let $R$ be a commutative ring and $M$ be an $R$-module. ...
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16 views

Under what condition, does this universal property of tensor algebra hold?

Let $R$ be a commutative ring and $M$ be an $R$-module. Note that the tensor algebra $T(M)$ is a unital associative $R$-algebra. Below is the universal property of the tensor algebra. Theorem: ...
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23 views

Universal property of tensor product of $R$-algebras

Let $R$ be a commutative ring and $A_1,...,A_{n+1}$ be $R$-algebras. Let $A_1\otimes_R\cdots\otimes_R A_{n+1}$ be equipped with the natural $R$-algebra structure. Let $N$ be an $R$-algebra. Let ...
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34 views

Matrix into a 4-tensor

I've been trying to figure out a way to add a dimension to a matrix with a certain rule: Let our matrix X have dimensions NxM, L be some positive number, and the desired resulting 4-tensor (3d ...
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1answer
41 views

How do I use Einstein notation to propose a rule for $MX=0$ and proving it's linear?

Let $$M = \begin{bmatrix}a^1_1 & a^1_2&....& a^1_k\\a^2_1 & a^2_2 & .... & a^2_k \\ : & : & .... &: \\ a^r_1 & a^r_2 & .... & a^r_k\end{bmatrix} ...