# Tagged Questions

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### How to write an analogue to matrix-vector multiplation with an extra dimension in tensor notation

My background is severely lacking in tensor algebra, and after a few days of looking into tensors I am still not able to even formulate this question quite correctly; my apologies for that. I am aware ...
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### On isomorphisms of tensors of certain type

I've got a question form Gille and Szamuely's "Central Simple Algebras' and it's about vector spaces equipped with tensors of certain types. Let $V$ be a $k$-vector space. For a field extension ...
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### Let $K$ a field, $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor. Prove that $f \wedge f =0$

Let $K$ a field, with $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor ($f \in {\mathcal T}_n(V):=\Lambda^{n}(V)$), i.e., $f$ is an multilinear ...
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### Tensor product and direct sums

I have an integral domain $R$, and $R$-modules $M$, $N_1$, $N_2$. I know that there is an $R$- module isomorphism $$M\otimes_R (N_1\oplus N_2)\cong (M\otimes_R N_1)\oplus(M\otimes_R N_2).$$ where ...
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### 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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### $\{u_{i} \otimes w_{j} \}_{i , j}$ forms a basis for $U \otimes W$

Suppose $U$ and $W$ are $k$-vector spaces with bases $\{u_{i}\}_{i=1}^{n}$ and $\{w_{j}\}_{j=1}^{m}$. How to prove that $\{u_{i} \otimes w_{j} \}_{i , j}$ forms a basis for $U \otimes W$ ?
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### Tensor Product definition (help with certain step)

I'm going over some notes I took from the blackboard, and reached a slight hitch. I thought that maybe someone could help. Let $E,F$ be vector spaces over a field $\mathbb K$. A tensor product of $E$ ...
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### Universal property definition from Greub's Multilinear Algebra

Hi I started studying Greub's multilinear algebra book and I found something very strange when he defines the tensor product of two vector spaces: He defines: [...] Let $E$ and $F$ be vector spaces ...
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### Facts about the tensor product

I'm proving some statements about density operators, and would like to use two things. The problem is, that I'm not entirely sure they hold. Let $V,\; W$ be finite dimensional, complex inner product ...
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### Quick question: tensor product and dual of vector space

Recall that for a finite dimensional vector space $V$ we have the natural isomorphism $\phi :V^{*} \otimes V \rightarrow Hom(V,V)$ given by $\alpha \otimes v \mapsto (x \mapsto \alpha (x)v)$. Is ...
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### For covariant tensors, why is it $\bigwedge^k(V)$, not $\bigwedge^k(V^*)$?

In learning the very basics of differential geometry, I have seen the exterior product defined a couple of ways: First, I have seen it as the image of the covariant tensors (which I believe are ...
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### Alternating tensors vs $p$-vectors

Is there a reason to differentiate between alternating tensors and $p$-vectors? More precisely, is the exterior algebra always isomorphic to the subalgebra of alternating tensors? Thanks.
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### The definition of addition on the tensor product of Hilbert spaces

Let $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ respectively. Construct the tensor product of $H_1$ and ...
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### Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$V\otimes W := L_2(V^*\times W^*,\Bbb F)$$ I am also aware that this space is isomorphic to the tensor ...
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### Operations on vector spaces

Is there a symmetrical analogue of the grassmann tensor products? Is it the polyadic tensor product? Are such symmetrical products used anywhere in differential geometry?
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### Dimension of Bil(V)

Let $V$ be a vector space of finite dimension $n$, and let $\operatorname{Bil}(V)$ be the vector space of all bilinear forms on $V$. In some notes by Keith Conrad, he says in an exercise that ...
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### Doubt in argument in proof of universal property

I have a doubt regarding one argument used to prove that the tensor product indeed has the universal property, namely that for any multilinear map $f:V_1\times\cdots\times V_p\to W$ if the tensor ...
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### Tensor Product universal property

Let $V,W$ be vector spaces and $T$ the following mapping: \begin{align*} T:V\times W&\to V\otimes W\\ (v,w)&\mapsto v\otimes w \end{align*} Then $(V\otimes W,T)$ Satisfies the universal ...
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### Tensor Products, various defintions

I came across a definition for the tensor product which differs from the standard definition. This book defined the tensor product of vector spaces $V$ and $W$ as the space $L(V,W,\Bbb K)$ of bilinear ...
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### Motivation for the Tensor Product

I've already asked about the definition of tensor product here and now I understand the steps of the construction. I'm just in doubt about the motivation to construct it in that way. Well, if all that ...
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### Graded Vector Spaces and the Interchange Law

I'm a little confused about how to correctly interchange factors in tensor products on graded vector spaces. In particular let $V:= \bigoplus_{n \in \mathbb{N}} V_n$ be a $\mathbb{N}$-graded vector ...
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### Kernel of the Tensor Product of a Linear Map with Itself

For two vector spaces, $V$ and $W$, and a map $f: V \to W$, it is clear that: $$\ker(f) \otimes V + V \otimes \ker(f) \subseteq \ker(f \otimes f).$$ Does the opposite inclusion hold? If so, I'd ...
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### What does “a map is isomorphic to another map” mean?

In this article, there are two proposition as the following: 1.(Proposition.) For every bilinear map $f:V\times W\to U$ there is a unique linear map $h:V\otimes W\to U$ such that $hg=f$, where $g$ ...
In Gowers's article "How to lose your fear of tensor products", he uses two ways to construct the tensor product of two vector spaces $V$ and $W$. The following are the two ways I understand: ...
### understanding of the tensor product $V^*\otimes W^*$
In S.S.Chern's Lectures on Differential Geometry, I don't understand the following text in Chapter 2, which introduces the tensor product: The tensor product $V^*\otimes W^*$ of the vector spaces ...