# Questions tagged [tensor-products]

For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.

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### $R$-module structure of tensor product of $R$-algebras (Atiyah-Macdonald)

I am (re-)reading the section on tensor products of $R$-algebras in Atiyah-Macdonald, where $R$ is a commutative ring, and I am not sure about the given definition of the $R$-module structure on the ...
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### How is it justified to take the tensor product of nonlinear functions?

Suppose $f, g: \mathbb{R}^n \rightarrow \mathbb{R}^n$. If $f$ and $g$ are both linear then we may define the tensor $$(f \otimes g)(x_1,\ldots,x_{2n}) = f(x_1,\ldots,x_n)g(x_{n+1}, \ldots,x_{2n})$$ by ...
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### Universal Property of tensor product of $\mathbb Z_2$-graded algebras.

If $A$ and $B$ are two $k$ algebra's with a $\mathbb{Z}_2$-grading then I know that a $\mathbb{Z}_2$-graded structure can be defined on their tensor product. One does this by altering the ...
1 vote
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### Understanding the order of tensor application as a linear map.

I have a question regarding the application of one tensor to another, $Q(D)$. Let's start with the simplest example, considering the bilinear form $G=G_{ij}(\epsilon^{i}\otimes \epsilon^{j})$ acting ...
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### Projective tensor product of operator spaces

Consider the following fragment from Effros and Ruan's book "Operator spaces" Why is a decomposition as in the red box possible? In fact, it is not even clear to me that any such ...
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### Computer algebra system for Tensor Algebra

I am searching for a computer algebra system that allows to tackle the following problem: Let $V = \mathbb{R}^3$. Consider the free commutative algebra $\mathrm{S}^\bullet V$, with $\mathrm{S}^kV$ ...
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### trace of wedge product and cyclic property [closed]

Surprisingly, while their are similar but more advanced questions on this site, I don't see any answers to the basic version I am asking herein. If I am taking the trace of a wedge product of matrices,...
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### Traceless tensor product: unclear definition

I do not understand here on the page $8$ that in the definition of traceless tensor product there are 3 indices $i,j,k$ in the rightmost term but in the preceding ...
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### R is torsion free is if and only if the natural map is injective

I'm reading Silverman's Arithmetic of Elliptic Curves which has the following statement: A ring R is torsion free is equivalent to saying, the map R$\longrightarrow$R$\otimes\mathbb{Q}$ is injective. ...
1 vote
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### Prove Tensor-hom adjunction via Hom-set definition

I am trying to learn some basics of category theory, precisely Adjunction, but I've encountered some difficulties trying to prove such a statement. So, I want to prove that tensor product $- \otimes X$...
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Let $R$ be a field of characteristic zero, and $M$ be a $\mathbb{Z}$-module. Knowing that $R\otimes M$ is a vector space, can we deduce that $M$ is a free $\mathbb{Z}$-module and $\text{rank}(M)=\... 0 votes 0 answers 26 views ### Does bases of operators on two Hilbert spaces span the operators on the tensor product of the two spaces? Let$\mathcal H_1, \mathcal H_1$be two Hilbert spaces,$\mathcal H=\mathcal H_1\otimes\mathcal H_2$is their tensor product. Let$\mathcal L(\mathcal H_1),\mathcal L(\mathcal H_2),\mathcal L(\mathcal ...
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Let $\vec x\in \mathbb{R}^n$, and $A\in\mathbb{R}^{m\times n}$. If I wanted to compute a linear map $$\vec x\mapsto A\vec x,$$ it would suffice to compute it on $n$ unit vectors $u_i = A\vec e_i$. ...