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

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1
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1answer
168 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
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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 ...
7
votes
6answers
146 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
85 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
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1answer
65 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
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1answer
55 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
64 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
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0answers
45 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
94 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
40 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
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0answers
80 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
91 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
373 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
80 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
34 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
38 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
103 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
75 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
104 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
16 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
26 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 ...
1
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1answer
83 views

Decompose $\omega:= e_0\wedge(e_1\wedge e_2 + e_3\wedge e_4)$

$(e_1\wedge e_2 + e_3\wedge e_4)$ is well-known to be of rank 2 (can't be decomposed). On the other hand, $\omega \wedge \omega = e_1\wedge e_1 \wedge ... = 0$. According to the wikipedia article ...
1
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1answer
53 views

Correct definition of bilinear(multilinear) maps over noncommutative rings

I wonder about the following: Let $R$ be a noncommutative ring with $\alpha, \beta \in R$ such that $\alpha \beta \neq \beta \alpha$. Let $M,N,P$ be left-$R$-modules. Why does it make problems to ...
0
votes
1answer
41 views

Do we need any special justification for concluding $f=0$?

Today, we were showing the following in class: Let $V$ be a finite-dimensional vector space over $F$. Prove that the bilinear map $$\beta:V \times V^* \to F, \ \ (v,f) \mapsto f(v)$$ is ...
1
vote
1answer
46 views

Showing $\det(I_E+f)=\sum_{p=0}^{n}\mathrm{tr}(f^p)$ where $f^p:Λ^{p}(E) \to Λ^{p}(E)$ and $f:E\to E$

$E$ is a vector space with dimension $n$ and $f:E\to E$ is a linear map and for every $p=1,2,3,...$ we have $f^p:Λ^{p}(E) \to Λ^{p}(E)$ which is defined as below ...
1
vote
2answers
52 views

$(n-1)$-alternative tensor on E are decomposable

$E$ is a real vector space with dimension $n$ and $E^*$ is dual space of $E$. Assume $\alpha \in Λ^{n-1}(E)$ Show that there exists $\alpha_1,\alpha_2,...,\alpha_{n-1} \in E^*$ such that ...
4
votes
1answer
82 views

Almost complex structure which fails to be compatible

Let $V$ be a real vector space equipped with a scalar product $\langle, \rangle$ (i.e. a positive definite symmetric bilinear form). We say that an endomorphism $J: V \to V$ is an almost complex ...
2
votes
2answers
98 views

How does the representation of co-vectors change if we change the basis of a vector space $V$?

I'm trying to understand how vectors, differential forms and multi-linear maps in general transform under change of coordinates. So I start with the simplest case of vectors. Here's my own attempt, ...
0
votes
1answer
25 views

symmetric bilinear form on $\mathbb{Z}_2$

Find all symmetric bilinear forms of a vector space $V$ of finite dimension on $\mathbb{Z}_2$. As every bilinear form is represented by a matrix then the idea is to find the set of matrices ...
1
vote
1answer
49 views

Is $\psi (x_1,…,x_n)=\det \begin {pmatrix} x^1_1 & \dots &x^1_n \\ \vdots & &\vdots \\ x^n_1 & \dots & x^n_n\end{pmatrix}$ multilinear?

Suppose $\forall x_1,\ldots,x_n,\in R^n$, denote $x_1=(x_1^1,x_1^2,\ldots,x_1^n),x_2=(x_2^1,\ldots,x^n_2)$. Define $\psi:V\times V\times \dots\times V\to R$ as follows: $$\psi (x_1,\ldots,x_n)=\det ...
0
votes
1answer
70 views

How to identifiy $V \wedge V$ with the space of all alternating bilinear forms

Let $\{ e_i \}$ be a basis for $V$, then the space of tensors $V \otimes V$ could be identified with the space of all formal sums $\sum_{ij} \alpha_{ij} (e_i, e_j)$ (I know a base independent approach ...
2
votes
0answers
68 views

Unnecessary Elements in the Tensor Product?

For vector spaces $U, V$ there exits a unique (up to isomorphism) vector space, denoted by $U \otimes V$, and a bilinear map $\eta : U \times V \to U \otimes V$ such that for every bilinear map $\xi : ...
0
votes
1answer
125 views

Different Definitions of Tensor product, Halmos, Formal Sums, Universal Property

In the classic Finite-Dimensional Vector Spaces by P. Halmos he defines the Tensor product as The tensor product $U \otimes V$ of two finite-dimensional vector spaces $U$ and $V$ (over the same ...
1
vote
1answer
53 views

Tensor analog of Matrix Product

Given two $n \times n$ matrices $A$ and $B$, we can form their matrix product in the usual way. Is there a similar product for tensors? E.g., if one is given two $n \times n \times n$ tensors ...
0
votes
1answer
30 views

Induced Action of Matrix on Tensor Product

I've been asked to write the induced action of a matrix in $M_4(\mathbb{R})$ on $\mathbb{R}^4 \otimes \mathbb R^4$, but this terminology is unfamiliar to me. What does it mean for a matrix to induce ...
0
votes
1answer
141 views

$G$-representations, $W \otimes V^* \to \text{Hom}(V,W)$

Let $V$ and $W$ be finite-dimensional vector spaces. I know how to construct an explicit isomorphism of vector spaces $W \otimes V^* \to \text{Hom}(V,W)$ and show that it's an isomorphism. But if I ...
1
vote
1answer
36 views

What is explicit isomorphic map between $\Bbb R$ and $\text{Alt}^n(\Bbb R^n)$

We denote $V \subset \mathbb{R}^n$ to be an open set. Definition 1: Let $\text{Alt}^p(\Bbb R^n)$ denote the set of all alternating multlinear functionals on $\Bbb R^n$. Definition 2: Let ...
0
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0answers
29 views

Multilinear Function proof in Spivak?

Note that $$\wedge^n (V)$$ denotes the set of all alternating multilinear functions and $\mathfrak{I}^n(R^n)$ denotes the set of all multilinear function. I don't know what the actual symbol ...
0
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0answers
35 views

A matrix equation with real coefficients

The problem is the following: Find $\lambda$ such that $ b^{T}A\left[A^{T}A-\lambda L^{T}L\right]^{-1}L^{T}L\left[A^{T}A-\lambda L^{T}L\right]^{-1}A^{T}b-\delta^{2}<0 $ where ...
2
votes
1answer
32 views

Multilinear algebra and matrices

Given $\wedge^k(V)$ an alternating multilinear space and $T : V \to W$ a linear map, then we have $$v_1 \wedge \dots \wedge v_k \in\wedge^k(V).$$ Define $$\wedge^k(T)(v_1\wedge\dots\wedge v_k) = ...
8
votes
1answer
214 views

How can I determine the number of wedge products of $1$-forms needed to express a $k$-form as a sum of such?

This question was motivated by this related one: How "far" a differential form is from an exterior product . Let $\mathbb{V}$ be a vector space of dimension $n$ with underlying field ...
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2answers
87 views

Help with a paper about tensors

I came across something in a paper I am not able to understand jet. Unfortunately the author is kind of short with explanations. Maybe someone here can help me to understand this. $M^d \in ...
0
votes
0answers
26 views

Indices question in this multilinear algebra question.

Suppose $V$ is finite dimensional with $\dim V = n$ and $f : V \to V$ be linear. Prove that there is a number $d(f)$ such that $$\Omega^n(f)(\omega) = d(f)\omega.$$ Here, $\Omega^n(V)$ denotes the ...
0
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2answers
73 views

Multilinear Algebra, finding $z \wedge z.$

Define $A^k(V)$ to be the set of all alternating multilinear functions from $V^k \to \mathbb{R}$. Consider the space $A^2(\mathbb{R}^4)$, does there exists $z\in A^2(\mathbb{R}^4)$ such that $z ...
0
votes
1answer
48 views

Question about determinant as transformation on alternating multilinear $n$-forms

If $T:E \to F$ is a linear transformation, $f : F\times\cdots\times F \rightarrow \mathbb{R}$ is an alternating, multilinear $n$-form and $\overline{T}:A_n(F)\rightarrow A_n(E)$ is a function that ...
1
vote
1answer
43 views

Polynomial and super-symmetric tensor

A quadratic function uniquely determines a symmetric matrix. Ok that’s easy. Now a homogeneous polynomial function $f(x)$ also uniquely determines a super-symmetric tensor. My question is how do I ...
0
votes
0answers
32 views

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 ...
0
votes
1answer
24 views

Intersection of two sets

Let $E\subset{\mathbb R}^n$ be a set of the type $I_1\times \dots \times I_n$, where $I_k$ are real intervals, and $X$ be and $n\times p$ real matrix. Suppose also that $rank(X)=p$ and $n>p$. Is ...
6
votes
0answers
128 views

Exploiting structure in multilinear equations

I'm wondering if there are any standard techniques for exploiting structure in multilinear equations. An example of what I have in mind is solving $A_{ab} X_{bc} A_{cd} (B_{ad} B_{bc} + B_{ac} ...
0
votes
0answers
30 views

Quotients in exterior products

I just started learning exterior products. The way I understand it, one can associate a subspace with with a bunch of spanning vectors using an alternating multilinear form. The 'k-blade' remains ...