# Tagged Questions

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

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### 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 vector space together with fixed symplectic bilinear ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\wedge$...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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\...
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### 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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### 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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### 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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### 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 ...
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### 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 ...
### 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 ...
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 ...