For questions about the extension of linear algebra to multilinear transformations of vector spaces.

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19
votes
4answers
457 views

Why the whole exterior algebra?

So, I've been reading up on multilinear algebra a bit. In particular, I've been reading up on the construction of of the exterior algebra of a finite dimensional vector space $X$, say over ...
4
votes
2answers
189 views

Basis for Tensor Product of Infinite Dimensional Vector Spaces

If V and W are vector spaces over a common field with bases $V_B = ${$v_i : i \in I$} and $W_B = ${$w_j : j \in J$}, then is {$v_i \otimes w_j: i \in I, j \in J$} a basis for $V \otimes W$ ? I have ...
0
votes
1answer
31 views

What is this map between tensor spaces called? (Change of coordinates).

Let $V$ and $W$ be finite dimensional vector spaces. Let $A: V \to W$ be linear. Define the map $A^* : (W^*)^{\otimes r} \to (V^*)^{\otimes r}$ by $(A^*\alpha)(w_1, w_2, \ldots, w_r) = \alpha(Aw_1, ...
0
votes
1answer
27 views

Define $f: V \times V \rightarrow \mathbb{K}$ ($\mathbb{K}$ field), by $f(u, v) = \langle u, Av\rangle$

Define $f: V \times V \rightarrow \mathbb{K}$ ($\mathbb{K}$ field), by $f(u, v) = \langle u, Av\rangle$, any $u$, $v \in V$ and $A \in M(n, \mathbb{R})$ with $A^T = -A$, i.e., $A$ is skew-symmetric. ...
2
votes
0answers
46 views

Natural Isomorphism between $V^*\otimes W^*$ and $\mathcal L^2(V,W; F)$.

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and ...
4
votes
1answer
65 views

Invariants of $V^{\otimes N}$. [closed]

Let $V$ be a finite dimensional complex vector space, and $G = SL(V)$ be the group of linear transformations of $V$ with determinant $1$. (a) Show that $V^{\otimes N}$ contains a nonzero ...
2
votes
1answer
53 views

why integrating only alternating forms?

Hello I was reviewing some concepts of differential forms. I cannot recall why only multilinear alternating forms can be integrated on manyfolds and not general multilinear forms... Why is the ...
1
vote
3answers
93 views

Why is quadratic form defined via a symmetric bilinear form?

A typical definition of quadratic form goes like this: Let $B:V\times V \to F$ be a symmetric bilinear form. A function $Q : V → F$ defined by $Q(v) = B(v, v)$ is called a quadratic form. Why ...
0
votes
1answer
84 views

What does it mean to say that determinant is multilinear?

Can someone clearly explain to me what is meant by the determinant being multilinear, and what (multi-)linear functions are? I can't find a clear answer to this question.
1
vote
1answer
28 views

Isomorphism between different definitions of symmetric tensors

Hi I was working through symmetric tensors (technically in a differential manifold, but we defined it abstractly). We defined them first in the usual way: Consider $A$ an integral domain and ...
3
votes
2answers
52 views

To show $L\otimes_K \text {End}_K(V)\cong \text{End}_L(L\otimes_K V)$

Let $L/K$ be field extension and let $V$ be a $K$-vector space. Then do we have an isomorphism $$L\otimes_K \text {End}_K(V)\cong \text{End}_L(L\otimes_K V)$$ as $L$-algebras? My attempt: for ...
0
votes
0answers
57 views

Introduction to tensor (for graph analysis)

I am starting a PhD program in social network analysis and I would like to have some suggestions about introductory books and online material to Tensor calculus. I am a complete newbie in algebra. ...
0
votes
0answers
30 views

Help in the proof of Poincaré Theorem to differential forms

I'm revising the proof of Poincaré Theorem, but I don't understand a pass of proof. Let be $E$ and $F$, Banach spaces and $U\subset E$ open set. Consider $\omega\in\Omega_p^n(U;F)$ a p-differential ...
0
votes
1answer
41 views

How to prove some formulations aboult Kronecker product?

The Kronecker product has some properties as the wikipedia http://en.wikipedia.org/wiki/Kronecker_product. For the sake of simplicity, we denote $\mathbf{U}_{M}^T\otimes ...
2
votes
0answers
84 views

Eigenvalues of a Kronecker Product type matrix

We have matrix $C$ of the form: $C =\begin{bmatrix} B_{1,1} A_{1,1} & B_{1,2} A_{1,2} & \dots & B_{1,K} A_{1,K} \\ B_{2,1} A_{2,1} & B_{2,2} A_{2,2} & \dots & B_{2,K} ...
2
votes
0answers
55 views

How to prove the formulation of mode-$n$ matricization and preclusive mode-$n$ product?

The mode-$n$ product of a tensor $\mathcal{X}=[x_{i_1,\ldots,i_M}]\in \mathbb{R}^{I_1\times \cdots \times I_M}$ and a matrix $\mathbf{U}=[u_{i_m,j}]\in \mathbb{R}^{I_m\times J}$ is denoted by ...
0
votes
0answers
37 views

Linear independence of differential 1-forms

Let be $(E,\mathbb{K}),(F,\mathbb{K})$ Banach space and $U\subset E$ open set. If $f_1,...,f_n\in\Omega_1(U;F)$ differential 1-forms are linearly independents, where $$\Omega_1(U;F)=\{f:U\rightarrow ...
1
vote
1answer
45 views

Show that the trace of the operator $S \wedge T$ is zero

I have some difficulties with the following problem: Let $V$ be a finite dimensional vector space over $\mathbb{K}$. Let $S,T \in L(V,V)$. Show that the trace of the operator $S \wedge ...
1
vote
2answers
70 views

Question about tensor products, decomposable tensors, …

I need some help with the following problem: Let $V_1,\ldots,V_m$ be finite dimensional vector spaces over $\mathbb{K}$. Let $\varphi \in L(V_1,\ldots,V_m;U)$ such that $Im(\varphi)=U$. ...
1
vote
1answer
30 views

Relationship with trace and asymptotic stability in control theory [closed]

What is the relationship between $\mathrm{tr}(\exp(tA) \exp(tA^\ast))$ and asymptotic stability in control theory ?
6
votes
1answer
91 views

Differentiation on Manifolds Basics

I'm having some real trouble comprehending integral curves and Lie derivatives on a Manifold. I will write out my understanding and ask the questions below. For a vector field $X$ on smooth manifold ...
1
vote
1answer
59 views

Tensor product definition?

I am getting a bit confused on the notation used for tensor products, is we have the tensor product space $V\otimes V^*$ if $v\in V$ and $a \in V^*$ then is the following correct? $$v \otimes ...
7
votes
1answer
132 views

Tensors as Multilinear maps?

Today I learned about Tensors as multilinear maps. I usually think of tensors as a multidimensional array of numbers with fixed transformation laws, and I am having trouble understanding how tensors ...
7
votes
1answer
205 views

Eigenvalues of Kronecker Product

Maybe it's simple but I can't see the solution of this problem (Russell Merris, Multilinear Algebra, CRC Press, 1997, chapter 6, p.202, exercise 4): Let $\lambda_1,\ldots,\lambda_p$ be the ...
1
vote
1answer
51 views

Defining addition of vectors of different dimensions

While doing real data analysis I came up with a problem. I have given lots of efforts to solve it and could not succeed. Here is the problem: Suppose, we have a set of vectors ...
2
votes
0answers
84 views

Finding a maximal isotropic subspace

I have the following question: Let $V$ be a finite dimensional complex vector space. For a given bilinear form $(,): V \times V \rightarrow \mathbb{C}$, a subspace $W$ of $V$ is called isotropic with ...
2
votes
1answer
59 views

Inverse of covariant tensor of rank two is contravariant.

I'm studying tensors on my own, using "Tensor Calculus" from David C. Kay, and there is this theorem in page $29$: Suppose that $(T_{ij})$ is a covariant tensor of order two. If the matrix ...
1
vote
1answer
29 views

Multilinear maps: is $\phi(av_1,v_2)$ always equal to $\phi(v_1,av_2)$?

I am learning about multilinear maps by myself and the book I'm following gives a definition which is somewhat vague. That's the definition: Given vector spaces $V_1,V_2,\dots,V_p,W$. A mapping ...
2
votes
1answer
61 views

Multilinear algebra some basics.

The wedge product of $p$ vectors in vector space $V$ is called a $p$-vector and the vector space generated by all $p$-vectors is denoted $\bigwedge^p V$ with the basis $e_I:=e_{i1}\wedge\dots\wedge ...
3
votes
2answers
516 views

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions?

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions? Needed for the irrep decompositon of $3\otimes 3\otimes 3$ in here. No idea where to start to ...
1
vote
1answer
62 views

acrobatics with $2$-form in $\mathbb{R}^{2n}$ [closed]

In the space $V = \mathbb{R}^{2n}$ with coordinates $(x_1, \dots, x_n, y_1, \dots, y_n)$ consider the $2$-form $\omega = \sum_{i=1}^n x_i \wedge y_i$. Let $A$ be a $n \times n$ matrix. Consider a ...
8
votes
6answers
166 views

coordinate free proof that $\text{div}(\nabla f \times \nabla g) = 0$

Let $V$ be a Euclidean $3$-dimensional space. Does there exist a coordinate-free proof that for any two $C^1$-functions $f, g: \mathbb{R}^3 \to \mathbb{R}$ we have $$\text{div}(\nabla f \times \nabla ...
6
votes
2answers
89 views

Does a $p$-form eat $p$-vectors or $p$ number of vectors?

A bilinear form is another term for a $2$-form. So does it eat $2$ distinct vectors or a single $2$-vector?
1
vote
1answer
70 views

Symmetric algebra

If $V$ is a vector space over the field $K$ with basis ${v_1, v_2,…,v_n}$, then the symmetric algebra $S(V)= K[v_1,v_2,..,v_n]$. The question is: If $K$ is a commutative ring, then this equality is ...
0
votes
1answer
59 views

The $n$-fold wedge product of a $2$ form

For the $2$-form $\omega$ on $\Bbb R^{2n}$, $$\omega = \sum_{i = 1}^{2n-1} dx_i \wedge dx_{i + 1}$$ why is $\bigwedge_{i = 1}^n \omega \neq 0$? I thought that if $n = 1$, (testing just $3$ terms) ...
2
votes
1answer
68 views

Künneth formula in topology, show isomorphism

Where could I find a proof of the isomorphism aspect of Theorem 2.4 in this pdf: http://math.stanford.edu/~conrad/diffgeomPage/handouts/tensor.pdf For vector spaces $V$ and $W$, consider $V$ and ...
1
vote
0answers
47 views

Can the gradient be expressed with contravariant components?

I read that the gradient is an example of a quantity that transforms covariantly since in the below expression for the gradient $$\frac{\partial x^j}{\partial x'^i}$$ appears instead of ...
2
votes
1answer
114 views

Components of vector in dual basis transform covariantly

I am trying to understand how components of a vector in the dual basis transform covariantly as mentioned in this quote. If you seek to define a quantity (such as vector A) that remains ...
3
votes
1answer
41 views

computation involving exterior $2$-form on $\mathbb{R}^n$

Let $$\theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1}$$be an exterior $2$-form on $\mathbb{R}^n$, and $A, B \in \mathbb{R}^n$ are vectors$$A = (1, 1, 1, \dots, 1),\text{ }B = (-1, 1, -1, \dots, ...
0
votes
0answers
81 views

Do orthogonal vectors yield orthogonal bivectors?

Suppose we have a (say, ordered) set $$(X_1, \ldots X_k)$$ of pairwise orthogonal vectors, say with span $S$. This determines a set $$(Y_{ij})_{1 \leq i < j \leq n}, \qquad Y_{ij} := X_i \wedge ...
1
vote
1answer
96 views

wedge product, multilinear algebra in $\mathbb{R}^{2n}$

Denote coordinates in the space $\mathbb{R}^{2n}$ by $(x_1, y_1, \dots, x_n, y_n)$. Consider a $2$-form $$\omega = \sum_{i=1}^n x_i \wedge y_i.$$ (a) Compute$$\underbrace{\omega \wedge \dots \wedge ...
9
votes
4answers
408 views

covariant and contravariant components and change of basis

I encountered the following in reading about covariant and contravariant: In those discussions, you may see words to the effect that covariant components transform in the same way as basis ...
5
votes
2answers
104 views

Determinant of exact sequence

Let $0 \to A \to B \to C \to 0$ be an exact sequence of vector spaces. I want to show that I have a canonical isomorphism $$\text{det}(B)= \text{det}(A) \otimes \text{det}(C).$$ Here, "det" refers ...
1
vote
2answers
36 views

exists a 1-form given exterior 2-form, 1-form on $3$-dimensional space?

Let $\alpha$ be an exterior $2$-form, and $\beta$ is a $1$-form on a $3$-dimensional space. Suppose that $\alpha \wedge \beta = 0$. How do I go about showing there exists a $1$-form $\gamma$ such that ...
2
votes
1answer
39 views

expression of the 4-tensor $f \otimes g$ in given basis

Let $f$ and $g$ be bilinear functions on $\mathbb{R}^n$ with matrices $a = \{a_{ij}\}$ and $B = \{b_{ij}\}$, respectively. How would I go about finding the expression of the $4$-tensor $f \otimes g$ ...
-2
votes
1answer
110 views

differential forms, cylindrical coordinates, geometric interpretation [closed]

Consider a differential $1$-form $\beta$ which in cylindrical coordinates $(r, \theta, z)$ has the form $$\beta = f(r)\,dz + g(r)\,d\theta,$$where $g'(0) = 0$. Find a condition when $\beta \wedge ...
3
votes
1answer
82 views

Multilinear form as scalar multiple of determinant function

While going through Hungerford's $\textit{Algebra}$, there is a theorem of linear algebra which states that every alternating $R$-multilinear form $f$ on $M_n(R), R$ a commutative ring, is a unique ...
2
votes
1answer
118 views

When are two simple tensors $m' \otimes n'$ and $m \otimes n$ equal? (tensor product over modules)

Suppose that $M$ is a right R-module and $N$ is a left $R$-module. We can construct $M \underset{R}\otimes N$ and give it an Abelian group structure by considering the free R-module $K$ generated by ...
0
votes
1answer
19 views

common factors of multilinear polynomial

Say $F,G\in\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ are two multilinear polynomial. If $F$ and $G$ vanish at a common set of coordinantes $(a_{i1},a_{i2},\dots,a_{in-1},a_{in})\in\Bbb R^n$ for ...
3
votes
0answers
27 views

On Schur map and tableaux

My post refers to Jerzy Weyman's "Cohomology of vector bundles and syzygies" pag. 37. Let $R$ be a ring and $E$ a free $R$-module of rank $n$. Let ${e_1, \cdots, e_n}$ a basis of $E$. Let us consider ...