0
votes
0answers
19 views

Exterior algebra of a direct sum [duplicate]

Why the exterior algebra of a direct sum of subspaces is isomorphic to a Tensorial product of it's exterior algebras, I mean, Why ⋀(V⊕W)≃⋀V⊗⋀W ?
0
votes
0answers
33 views

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 ...
0
votes
1answer
48 views

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 ...
2
votes
1answer
50 views

“Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions

Is there some precise sense in which the "alternation" functor $A$ that maps a multilinear function $f\colon M^d\to N$ to the alternating multilinear function $A(f)\colon M^d\to N$ defined by $$ ...
2
votes
4answers
190 views

Tensor product in multilinear algebra

In the book by Halmos ($FDVS$) the tensor product of two vector spaces U and V is defined as the dual of the vector space of all the bilinear forms on the direct sum of U and V. Is there a generalised ...
1
vote
2answers
57 views

Graded tensor algebra

Given a finite dimensional $\mathbb R$-vectorspace $V$ we can make $$ T(V) := \bigoplus_{n=0}^\infty V^{\otimes n}. $$ Here $V^{\otimes n} = V \otimes \cdots \otimes V$. An element of $T(V)$ looks ...
1
vote
1answer
47 views

$(V^*)^{\otimes n} \cong (V^{\otimes n})^*$

We assume that $V$ is finite dimensional. Make $\theta: (V^*)^n\to (V^{\otimes n})^*$ by $$ \theta(\alpha_1,\cdots,\alpha_n)(v_1 \otimes \cdots \otimes v_n ) := \prod_{i=1}^n \alpha_i(v_i). $$ Then, ...
1
vote
1answer
44 views

Exercise from Rotman, Advanced modern algebra , $\wedge^2 (\mathbb{Z}_p \oplus \mathbb{Z}_p) \neq 0$

(i) Let $p$ be a prime,. Show that $\wedge^2 (\mathbb{Z}_p \oplus \mathbb{Z}_p) \neq 0$ , where $\mathbb{Z}_p \oplus \mathbb{Z}_p$ si viewed as $\mathbb{Z}$-module ( with $\mathbb{Z}_p $ I mean ...
1
vote
1answer
31 views

Unclear Construction of Basis for Tensor Product

My problem lies in page 363 of Steven Roman's Advanced Linear Algebra (Here's a link). The author says that for each ordered pair $(e_i,f_j)$ where $\left\{e_i\right\}_{i\in ...
0
votes
1answer
22 views

wedge product - distributivity over addition

Wedge and tensor algebra are very new concepts to me and I want to understand how to prove the following property of the wedge product: ...
2
votes
1answer
43 views

wedge product and tensor operation problem

Let $\{e_{1},\ldots,e_{n}\}$ be the usual basis for $\mathbb{R}^{n}$ and let $\{\varphi_{i},\ldots,\varphi_{n}\}$ be the dual basis. Show that $$\varphi_{i_{1}}\wedge\cdots\wedge ...
3
votes
1answer
76 views

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 ...
0
votes
4answers
68 views

$\{u_{i} \otimes w_{j} \}_{i , j}$ 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 $U \otimes W$ ?
2
votes
1answer
54 views

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 $E$ ...
0
votes
1answer
76 views

Tensor product and valence of a tensor?

Given a vector space $V$, its dual vector space $V^{*}$ and a tensor $\mathbf{T}$: $\mathbf{T} \in \underbrace{V \otimes\dots\otimes V}_{n\text{ copies}}\otimes \underbrace{V^{*}\otimes\dots\otimes ...
0
votes
0answers
44 views

Tensor power of a quotient space

Let $E$ and $F$ be vector spaces, $F$ a subspace of $E$. Is there any canonical isomorphism between $E/F \otimes E/F$ and a quotient of the form $E \otimes E/G$, where $G$ is a subspace of $E \otimes ...
2
votes
1answer
51 views

Tensor Product and Direct Sum

Let $R$ be a commutative ring with identity and let $\{M_\alpha\}$ be a family of $R$-modules and $N$ another $R$-module. I've tried to show that $$\left(\bigoplus_\alpha M_\alpha\right)\otimes ...
2
votes
0answers
90 views

Recognizing pure tensors in tensor product of vector spaces

Let $V$ be a vector space and let $\{e_i\}$ be a basis for it. Then $\{e_I\equiv e_{i_1}\otimes...\otimes e_{i_r}\}$ is a basis for $V\otimes ... \otimes V$. Suppose I am given an element $w=\sum a_I ...
3
votes
1answer
66 views

Existence of isomorphism between tensor products.

In multilinear algebra many maps are usually proven to exist rather than simply defined. For example, commutativity is one such example. In the book I'm studying the author says: let $V_1,\dots,V_k$ ...
4
votes
2answers
131 views

Tensor Multiplication - Why should we use permutations?

I'm reading a book on multilinear algebra, and the author first establishes this easy isomorphism: if $V_1,\dots,V_k$ are vector spaces over the field $K$ and if $\sigma\in S_k$, then there is an ...
2
votes
3answers
111 views

How to really understand the tensor algebra?

If $V$ is a vector space over $F$, then we define $T^r_0(V)=V^{\otimes r}$, then we define the algebra of contravariant tensors to be $$T(V)=\bigoplus_{r=0}^\infty T^r_0(V)$$ together with the ...
2
votes
2answers
124 views

Eisenbud's proof of right-exactness of the exterior algebra

I'm trying to understand the proof in Eisenbud's Commutative Algebra that, given a right exact sequence $$K \to N \to M \to 0$$ of $R$-modules, we have an exact sequence $$K \otimes \wedge N \to ...
4
votes
1answer
212 views

What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity), then the tensor product $f_1\otimes ...
0
votes
1answer
48 views

Determinant of symmetric matrix of the form $v\otimes v$

Note that for $V=\mathbf{R}^n$, $$S^2V = \{ v\otimes w \mid v, w\in V\text{ and }v\otimes w=w\otimes v \} =\{ A\in \mathrm{M}_2(\mathbf{R}) \mid A=A^T \}.$$ Clearly, $S^2V $ contains $O=\{ v\otimes v ...
2
votes
1answer
74 views

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 ...
1
vote
1answer
78 views

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 ...
7
votes
1answer
267 views

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 ...
4
votes
2answers
146 views

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 ...
3
votes
1answer
149 views

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.
3
votes
1answer
95 views

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 ...
4
votes
1answer
248 views

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 ...
1
vote
0answers
59 views

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?
2
votes
1answer
29 views

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 ...
4
votes
2answers
120 views

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 ...
3
votes
1answer
193 views

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 ...
0
votes
0answers
67 views

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 ...
5
votes
5answers
1k views

Motivation for the Tensor Product [duplicate]

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 ...
1
vote
1answer
69 views

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 ...
5
votes
1answer
281 views

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 ...
2
votes
1answer
221 views

Tensor Algebra of Tensor Algebra

Suppose $V$ is a vector space and $T(V)$ the tensor algebra of $V$. What happens if we take $T(T(V))$ that is the tensor algebra of the (vector space) $T(V)$? I 'guess' I heard that $T(T(V)) \simeq ...
0
votes
1answer
64 views

Tensor Product Question

For a finite dimensional vector space $V$, is it true that $\bigwedge^{n - 1}V \otimes V = \bigwedge^{n}V \oplus \ker(\bigwedge^{n - 1}V \otimes V \overset{\psi}{\rightarrow}\bigwedge^{n}V)$ where ...
0
votes
0answers
74 views

Sum of series involving projectors on eigenspaces of Hamming association scheme

I am trying to understand this paper but I am stuck at this step. Can someone please help? Let $J_q$ be the $q\times q$ all ones matrix. Consider an orthonormal eigenbasis ...
7
votes
1answer
558 views

Condition for a tensor to be decomposable

Let $V$ be a vector space of dimension 3 with basis $e_1,e_2,e_3$. Let $W$ be a vector space of dimension 2 with basis $f_1,f_2$. Is $e_1\otimes f_1+e_2\otimes f_2$ decomposable? What about ...
2
votes
2answers
90 views

How can one see that $\operatorname{tr}(f\otimes g)=\operatorname{tr}f\operatorname{ tr }g$?

Suppose you have two free modules $M$ and $N$ of finite rank over a commutative ring $R$. Let's also take some $f\in\operatorname{End}_R(M)$ and $g\in\operatorname{End}_R(N)$, which gives a ...
3
votes
1answer
94 views

Why is $M_A\otimes_A N\cong M\otimes_R N$?

I've been doing some tensoring, but am having a hard time understanding the following isomorphism. Suppose $A$ is a commutative $R$-algebra, and for any $R$-module $M$, denote by $M_A=A\otimes_R M$. ...
7
votes
2answers
315 views

Tensor Decomposition

Consider a tensor product $$ V^{\otimes n} = \underbrace{V\otimes\cdots\otimes V}_{n} $$ where $V$ is a vector space over $\mathbb R$, $\dim V = m$ , hence $\dim V^{\otimes n} = m^n$ . So every $A ...
3
votes
1answer
333 views

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 $g$ ...
13
votes
4answers
3k views

understanding of the “tensor product of vector spaces”

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: ...
3
votes
1answer
468 views

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 ...