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

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Proving that for every $2$-form $\omega$, a basis of $V^*$ exists, so that $\omega$ can be written in a certain way

Let $\omega$ be an alternating $2$-form on an $n$-dimensional $\mathbb{R}$ vector space $V$, where $\omega$ isn't $0$ everywhere. I want to prove: There exists a basis $(\alpha_i)_{1 ≤ i ≤ n}$ of ...
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4answers
27 views

A Particular Symmetric Bilinear Map

Does there exist symmetric bilinear map on $\mathbb{C}^n$ such that $$<v, v> \neq 0 \quad \forall v \in \mathbb{C}^n?$$
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0answers
12 views

Represent of multilinear function (map)

$$ f:R^{k_1}\times ...\times R^{k_n} \rightarrow R $$ is a $n$ multilinear function , $k_i$ is positive integer.Then $f$ must can be represented as $$ f(x_1...x_n)=C\prod\limits_{i=1}^n<x_i,u_i> ...
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1answer
18 views

How to show $L(R,R;R)\cong R$?

$$ L(R,R;R)=\{f:R\times R\rightarrow R : f \text{ is bilinear } \} $$ Adding the addition and scalar product, $L(R,R;R)$ is a vector space . How to show $L(R,R;R)\cong R$ ? Besides what is the ...
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0answers
55 views

Geometric Intuition about the relation between Clifford Algebra and Exterior Algebra

It is common to see a relation being established between the Clifford Algebra and the Exterior Algebra of a vector space. Recently reading some texts written by Physicists I've seem applications of ...
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1answer
55 views

Deciding whether a form in the exterior power $\bigwedge^k V$ is decomposable

Let $V$ be a vector space and $\bigwedge^kV$ be the $k$th exterior power. I'm trying to find a condition that characterizes when an element $\omega \in \bigwedge^kV$ is decomposable in the sense that ...
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1answer
37 views

Show that a k-form can be expressed as wedge product

I'm trying to show that given a $1$-form $\omega$ and a $k$-form $\alpha$ such that $\alpha \wedge \omega = 0$ then there exists a $(k-1)$-form $\beta$ such that $\alpha = \omega \wedge \beta$. I'm ...
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0answers
12 views

CP decomposition representation for a tensor

Consider a third order tensor $X\in \mathbb{R}^{n_1\times n_2\times n_3}$, whose CP decomposition is $$X = \sum_{i=1}^r \sigma_i u_i \otimes v_i \otimes w_i, $$ where $\sigma_1\ge \sigma_2 \ge \cdots ...
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0answers
14 views

decomposable p-forms and linear independence

So I am attempting to show that: $$ z_1 \wedge \cdots \wedge z_p = 0 \implies z_1,\ldots,z_p \text{ linearly dependent} $$ my approach is to use the fact that: $$ z_1 \wedge \cdots \wedge z_p = \left[ ...
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0answers
23 views

Sesquilinear Forms

I was trying to solve some exercises related to sesquilinear forms: Let V be a C-vector space (C - complex numbers) Prove that the set $\mathcal{S}(V)$ of sesquilinear forms on V is a vector ...
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0answers
36 views

symmetric and alternating tensors in differential geometry

The following is an excerpt from Chern's Lectures on Differential Geometry: I don't see how the proof shows the other direction of the set inclusion. Would anybody explain the logic in the ...
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1answer
20 views

When do exterior and tensor algebras commute with dual spaces?

Suppose $V$ is a vector space, and $V^*$ is its dual space. Furthermore, let $\Lambda(V)$ be the exterior algebra of $V$, and let $T(V)$ be the tensor algebra. When do the following two statements ...
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0answers
21 views

What is the motivation for wanting to remove redundant terms that arise from the wedge product of two multilinear functions?

I am currently working through Loring Tu's "An Introduction to Manifolds," specifically the section in which the exterior algebra of multicovectors is introduced, and I am having trouble grasping the ...
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1answer
19 views

Does this symmetric rank-3 tensor vanish?

Suppose we have a rank-3 tensor $T$ on some vector space $\mathbb{V}$. We can view $T$ as a map: $$T: \mathbb{V} \times \mathbb{V} \times \mathbb{V} \to \mathbb{R},$$ which maps triples of vectors ...
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0answers
16 views

Exterior Powers of finite abelian group

Let $A$ be a finite $\mathbb{Z}$-module (i.e., a finite abelian group). My question is: for what $n\in \mathbb{Z}^{n\geq 2}$ the map \begin{align} \alpha_{n}:\bigwedge^nA&\to A^{\otimes n}\\ ...
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2answers
83 views

Doubt in definition of symmetric continuous function and norm in Kupka's paper

In this article "Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds" of I. Kupka has the following passage: For $H=l^{2}$ "Let $H^{*}$ be the dual of $H$. A base ...
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1answer
53 views

Linear algebra prerequisites for abstract algebraic geometry

I'm interested in what linear/multilinear algebra does one need to study algebraic geometry(following EGA and Harthshorne). Texts I have in mind are like "Foundations of algebraic geometry" by Ravi ...
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1answer
45 views

Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?

Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, and consider the $l_p$ norm ($p\geq 2$). Can we prove that the following problems are equivalent: $$\max_{\|x\|_p=\|y\|_p=1} \left| \langle x, ...
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0answers
22 views

Exterior power of a free module [duplicate]

Suppose $M$ is a free R module where $R$ is commutative and unital. Is $\Lambda^n M$, the nth exterior power of $M$ free?
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0answers
37 views

Higher order singular value decomposition

Can anyone give a clear explanation with example of the higher order singular value decomposition (HOSVD). All the references are theoretical stuff. I would like to have a simple example.
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0answers
38 views

Geometric interpretation of the determinant of a complex matrix

A complex $n$-dimensional vector space $V$ can be thought of as a real $2n$-dimensional vector space equipped with a map $J:V \to V$ with $J^2 = -I$. Complex-linear maps are then linear maps $V \to V$ ...
3
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0answers
57 views

Understanding the construction of Exterior Algebra

Background The tensor space of type $(r,s)$ associated with $V$ is the vector space $$\underbrace{V\otimes \ldots \otimes V}_{\text{r copies}} \otimes \underbrace{V^* \otimes \ldots \otimes ...
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1answer
31 views

linear algebra matrices and matrix operations

Show that if a square matrix $A$ satisfies $A^3 + 4A^2 -2A +7I = O$, then so does $A^T$.
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2answers
84 views

Assign a unique number to every linear form $\varphi : \mathbb R^n \to \mathbb R$, has this number a geometric interpretation?

If $V$ is a finite-dimensional vector space of dimension $n$, denote the space of all alternating $k$-fold multilinear maps (also called alternating $k$-tensors) by $\Lambda^k(V)$. Then $\dim ...
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1answer
53 views

Anisotropic scaling in geometric/Clifford algebra

Take the geometric algebra over $\Bbb R^n$. Suppose we have a blade multivector in this algebra. Now we want to anisotropically scale this multivector. Is there a general closed-form expression for ...
5
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1answer
23 views

Associating a $m-1$-tensor on $\mathbb R^m$ to an element of $\mathbb R^m$.

Show that for every alternating $m-1$-tensor on $\mathbb R^m$ there exists a unique $v \in \mathbb R^m$ such that for every linear function from $\mathbb R^m$ into $\mathbb R$ and for every ...
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0answers
21 views

The determinant of $X_I$ = wedge product of elementary tensors

Let $(x_1,\dots,x_k)$ be vectors in $\mathbb{R}^n$; let $X$ be the matrix $X = [x_1 \cdots x_k]$. If $I = (i_1, \dots, i_k)$ is an arbitrary $k$-tuple from the set $\{1,\dots, n\}$, show that ...
0
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1answer
29 views

Show that $T^*f$ is an alternating tensor if f is an alternating tensor

Show that if T: V $\rightarrow $ W is a linear transformation and if f $\in A^k(W) $ then $T^*f \in A^k(V)$ where $T^*$ is the dual transformation. Attept at the solution: If $f \in A^k(V)$ and ...
4
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1answer
89 views

In which sense is composition a tensor product

Let $\Phi\colon U\to V$ and $\Psi\colon V \to W$ be linear operators, and consider their composition $$ \Psi\circ \Phi $$ The operation, $$\circ:\mathcal{L}(U,V)\times\mathcal{L}(V,W)\to ...
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1answer
14 views

inequality in compressed sensing

Let $h\in R^n$ is a k-sparse vector, then how can i prove this inequality $$||h||_p\leq k^{1/p-1/q}||h||_q\ \ ,\forall p\in[1,q]$$ where $q\geq 1.$ please help.
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0answers
22 views

How do I find out if a multivariate function is multimodal?

I have programed a function in Matlab. Trying to optimize it, the local gradient based optimizer fails, while a global one doesn't. I suspect that the function is multimodal and that this is the cause ...
4
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0answers
65 views

Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space

I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces". We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the ...
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1answer
28 views

How to see directly that $A^*(V) \cong A(V)^*$?

Let $V$ be a vector space of finite dimension $n$, say over the field of real numbers. Now, I am aware that there is a canonical isomorphism $A^k(V^*) \cong A^k(V)^*$ between the space of alternating ...
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1answer
44 views

If and only if criterion for something to be a differential ideal

Let $I \subset \Omega^*(M)$ be a ($2$-sided) ideal (i.e. $I$ is a vector subspace, and for any $\alpha \in I$ and $\omega \in \Omega^*(M)$ we have $\omega \wedge \alpha \in I$). We say $I$ is a ...
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1answer
27 views

Validity of proof the continuity of the differentiation of the continuous bilinear form

Let $E, F, G$ be $3$ normed vector spaces. $B : E \times F \to G$ is a continuous bilinear form. Show that $B$ is differentiable on $E \times F$. I need to show that: $dB_{(a_1,a_2)}(v_1,v_2) = ...
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1answer
20 views

Wedge product is non-degenerate

Let $V$ be a k-dimensional vector space and $m,n \in \mathbb{Z}$ are such that $m+n \le k$. We consider the wedge product: $\wedge: \Lambda^nV \times \Lambda^mV \to \Lambda^{m+n}V$ defined by $(v,w) ...
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0answers
15 views

Differentiation of a continuous bilinear form

Let $E_1, E_2$ and $F$ be $3$ Banach spaces. Let $B: E = E_1 \times E_2 \to F$ be a continuous bilinear form. Show that $B$ is differentiable at every point $a = (a_1,a_2) \in E$ and its differential ...
2
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1answer
46 views

Definition of tensors

I'm studying the book "Riemannian Geometry" by Petersen and since I'm new to the subject, I'm helping myself also with the more introductory DoCarmos's book. I'm a bit confused about the definition ...
0
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1answer
39 views

Prove there exist such that $A = [T]_\beta$ and $B = [T]_\gamma$ [closed]

Let $A$ and $B$ be similar $n \times n$ matrices. Prove that there exists an $n$-dimensional vector space $V$, a linear operator $T$ on $V$, and ordered bases $\beta$ and $\gamma$ for $V$ such that $A ...
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1answer
35 views

About non-degenerate skew symetric form

The definition of a non-degenerate skew symetric $\omega : H \otimes V \to H^{*} \otimes V^{*} $, where H and V are finite dimensional vector spaces, is that for each $v \in V$ non-zero, $\omega : H ...
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0answers
23 views

Computation of a linear transformation of symmetric algebras

The following is a problem and my attempt at a solution. I would appreciate any further guidance on intuition or neat tricks involving this problem and related concepts. Also, where can I read more ...
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1answer
36 views

Showing that the tensor product of vector spaces is closed under addition

Let $V$ and $W$ be vector spaces over the field $F$. We know that $V\otimes W$ is a vector space over $F$. But how do we show closure under addition? For instance let $(a\otimes b), (c\otimes d)\in ...
2
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1answer
38 views

Proposition of a bounded set $A \subset \mathbb{R}^d$

I need some help continuing the proof for the following proposition. A set $A \subset \mathbb{R}^d$ is bounded $\Leftrightarrow$ there exists a cube $Q_s=[-s,s]^d=\left\{ (x_1,\dots,x_d) \: : \: ...
3
votes
1answer
26 views

Condition for nullity of quadrilinear form

I have been told the following. Lemma Suppose $V$ is a vector space over a field $K$, and $T:V\times V\times V\times V\to K$ is a multilinear map with the following properties holding for all ...
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1answer
43 views

Determinant of a tuple of vectors: is this a thing? If so, where can I learn more?

Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers. Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such ...
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2answers
47 views

Why are vectors considered to be rank (0,1) tensors and dual vectors considered to be rank (1,0) tensors?

Sean Carrol in his book of general relativity, he defines a tensor to be a multilinear map from a collection of dual vectors and vectors to $\mathbb{R}$: $T:T^*_p \times...\times T^*_p \times T_p ...
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1answer
36 views

Multilinear Mappings

Let $E$, $F$ complex Banach spaces and $p,q\in \mathbb{N}$ with $p+q\geq 1$. I will denote by $\mathcal{L}_a(^{p,q}E;F)$ the subspace of all $(p+q)$-linear mappings $A\in ...
4
votes
1answer
123 views

Trace of the $k$-th Exterior Power of a Linear Operator

Let $V$ be an $n$ dimensional vector space over a field $F$ and $T$ be a linear operator over $V$. Assume that the characteristic of $F$ is not $2$. Definition. Consider the map $f_1:V^n\to ...
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0answers
15 views

Cannonical forms for symmetric tensors of type (2,1)

Let $$T:V\times V \rightarrow V$$ be a symmetric (2,1) tensor, that is $$ T(X,Y)=T(Y,X) \,\ \forall X,Y \in V.$$ In the simple case when $\dim V=2,$ there is always a frame $F$ of $V$ , $F=(e_1,e_2)$ ...
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1answer
28 views

Universal Linearizer of Alternating Multi-$F[x]$-Linear Maps is Same as that of Multi-$F$-Linear Maps.

Let $V$ be a an $n$-dimensional vector space over a field $F$. Let $M=F[x]\otimes_F V$. We can consider $M$ as an $F[x]$-module by extending scalars using the inclusion $F\to F[x]$. Fact 1. There is ...