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

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0
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3answers
42 views

How to prove that $\phi(v\otimes f) = g(v,f)$ is injective?

Let $V$ a finite vector space and $V^{\ast}$ its dual. Let $g : V\times V^{\ast} \to \mathcal{L}(V,V)$ a bilinear map defined as follows: $$g(v,f)(w) := f(w)v.$$ To show that the map $$\phi(v\...
2
votes
2answers
46 views

How to show using the universal property that $V\otimes V^{\ast} \cong \mathcal{L}(V,V)$?

Let $V$ a vector space of finite dimension and $V^{\ast}$ its dual space. How to use the universal property to show that $V\otimes V^{\ast} \cong \mathcal{L}(V,V)?$ I just know that I can construct ...
0
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0answers
19 views

Infinitesimal Strain Tensor in a Cubic Crystal

I'm currently working through Vectors and Tensor in Engineering and Physics, and there's a problem regarding the strain tensor that I'm having a bit of trouble with. Given a cubic crystal with zero ...
3
votes
0answers
71 views

Canonical bundle of the Lagrangian Grassmannian

I'd like to compute the canonical bundle of the Lagrangian Grassmannian $\mathbb{LG}_n$, the set of Lagrangian subspaces of dimension $n$ of a complex vector space together with fixed symplectic ...
2
votes
2answers
35 views

Is it possible to define the tensor product of two vectors with respect to a bilinear form?

Given two vectors $\vec{v},\vec{w} \in \mathbb{R}^n$, and a bilinear form $\mathcal{B}$ represented by an $n \times n$ matrix $B$, we can define the inner product of $\vec{v}$ and $\vec{w}$ with ...
3
votes
2answers
44 views

generalization of positive-definite matrices to matrices over finite fields

Let $\mathbb{F}$ be a field, $\mathbb{F}^n$ be the $n$-dimensional vector space over $\mathbb{F}$, and $M_{n\times n}(\mathbb{F})$ be the space of $n\times n$ matrices with entries in $\mathbb{F}$. We ...
2
votes
1answer
35 views

Exterior algebra of a ring

In the book "Cohen-Macaulay rings" by Bruns and Herzog, the quick introduction of tensor algebra and exterior algebra left me a bit bewildered. After referring to the section on tensor algebra ...
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0answers
16 views

Linear subspaces of $\Lambda^2\mathbb R^5$

Let $\Lambda^2\mathbb R^5$ be the space of 2-vectors of $\mathbb R^5$. What are the linear subspaces $V$ of $\Lambda^2\mathbb R^5$ such that any element of $V$ is a simple 2-vector? Same question for ...
4
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1answer
49 views

How to represent matrix multiplication in tensor algebra?

How can we represent matrix multiplication in tensor algebra? Even if we assume all matrices represent contravariant tensors only, clearly matrix multiplication does not correspond to the ...
5
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0answers
69 views

Demystifying the tensor product

It seems to me, through my mathematical immaturity, that the tensor product seems to beg for more well-definition. I am working in vector spaces (so we always have a free module) and here is what my ...
0
votes
1answer
54 views

What exactly are operations involving tensors… In terms of their indices

So I have heard that tensor operations involve the faces of the rectangular prism. These are matrices right, and different properties of those matrices say things about the tensor? Could someone ...
0
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0answers
40 views

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra?

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra? https://en.m.wikipedia.org/wiki/Hodge_dual https://en.m.wikipedia.org/wiki/*-algebra This would seem to be a ...
0
votes
2answers
181 views

What is the “taxonomy” or “hierarchy” (partial ordering) of algebraic objects used to attempt to capture geometric intuition? [closed]

What follows is a list of terms all of whose relationships to one another I have never fully succeeded in establishing, despite having spent much of 6-8 years trying to so. Feel no need to give ...
1
vote
1answer
36 views

Let $ \omega = dx_1 \wedge dx_2 + dx_3 \wedge dx_4 + \cdots + dx_{2n-1} \wedge dx_{2n} \in \mathbb{R}^{2n}$. Find $\omega^n$ (in respect to $\wedge$) [duplicate]

Let $ \omega = dx_1 \wedge dx_2 + dx_3 \wedge dx_4 + \cdots + dx_{2n-1} \wedge dx_{2n} \in \mathbb{R}^{2n}$. Find $\omega^{n}$ (in respect to $\wedge$) When I say "$\omega^{n}$ (in respect to $\wedge$...
5
votes
1answer
113 views

Is there a commutative ring with a “generalized determinant”?

Does there exist a commutative ring(-with-a-1) $R$ and positive integer $n$ and function $\hspace{.04 in}f$ from [the set of $n$-by-$n$ matrices over $R$] to $R$ such that $f$ is linear in each row ...
0
votes
0answers
37 views

coordinate functions relation with covectors

I think that this should not be a difficult question to answer but I couldn't solve it by myself, so here is the question: Let $f_{1}, ..., f_{r}$ be $C^{\infty}$ functions on an open set $U$ of a ...
2
votes
2answers
47 views

Prove bilinear form is nondegenerate

This is from Linear Algebra, an Introductory Approach - Charles Curtis Let $V$ be a finitely generated vector space over $F$ with basis $\{v_1, ..., v_n\}$. Let $A = (\alpha_{ij})$ be a fixed $n$ by $...
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0answers
39 views

Does the “Leibniz multicategory over $R$” have an accepted name?

Let $R$ denote a commutative ring. Definition. The "Leibniz multicategory" over $R$ is given as follows: Objects. $R[D]$-modules (where $D$ is a formal symbol; an 'indeterminate'). ...
1
vote
1answer
23 views

Why this map implies the decomposition of the tensor product space?

Let $V$ be a vector space and consider the tensor product of $V$ with itself, that is $V\otimes V$. Define $\alpha' : V\times V\to V\otimes V$ by $$\alpha'(v,w)=w\otimes v.$$ In that case, $\alpha'$ ...
3
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2answers
39 views

Wedge product is nondegenerate symmetric bilinear form

Let$$f: \Lambda^k(\mathbb{R}^n) \times \Lambda^{n - k}(\mathbb{R}^n) \to \mathbb{R}, \quad f(\alpha, \beta) = \alpha \wedge \beta.$$How do I see that $f$ is a nondegenerate symmetric bilinear form?
0
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0answers
15 views

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 $V^*...
2
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5answers
47 views

A Particular Symmetric Bilinear Map

Does there exist symmetric bilinear map on $\mathbb{C}^n$ such that $$<v, v> \neq 0 \quad \forall v \in \mathbb{C}^n-\{0\}?$$
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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> ...
0
votes
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 ...
6
votes
1answer
86 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 ...
6
votes
1answer
63 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 $...
2
votes
1answer
39 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 ...
0
votes
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[ ...
0
votes
0answers
32 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 ...
1
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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 "...
1
vote
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 ...
0
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0answers
22 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 ...
2
votes
1answer
20 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 in ...
1
vote
0answers
17 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}\\ a_1\...
3
votes
2answers
84 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 ...
0
votes
1answer
63 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 ...
0
votes
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, ...
0
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0answers
23 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?
2
votes
0answers
38 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
40 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
62 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 V^*}_{\...
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1answer
32 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 $A^T$.
0
votes
2answers
86 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 \Lambda^...
0
votes
1answer
56 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
votes
1answer
24 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 $v_1, ...
1
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0answers
22 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 $\phi_{...
0
votes
1answer
36 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 $v_1,....
4
votes
1answer
95 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 \mathcal{L}(...
1
vote
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.
0
votes
0answers
23 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 ...