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

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Inference of an identity in Grassmann algebra.

I am reading Herbert Federer's book called "Geometric Measure Theory", in chapter one of Grassmann algebra, on pages 36-37, he says that for $f$ being an endomorphism of a finite dimensional inner ...
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What do we call the result of wedging together the columns of a matrix?

We can wedge together the columns of a square matrix to compute its determinant. More generally, the exterior product of the columns of a $b \times a$ matrix tells us the determinant of each $a \times ...
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+50

Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = ...
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1answer
33 views

Orbit closures of real symmetric bilinear form

Let $\alpha$ and $\beta$ be two real symmetric bilinear forms in $\operatorname{sym}(\mathbb{R}^n)$, with signatures $(p_{\alpha},n_{\alpha},z_{\alpha})$ and $(p_{\beta},n_{\beta},z_{\beta})$. I ...
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1answer
25 views

Alternating multilinear function satisfies $f(A)=\det(A)f(Id)$

I've just seen a proof of the statement: "Given $\alpha$ in a commutative ring $K$ there is a unique alternating multilinear function $f$ with $f(Id)=\alpha$." The determinant is defined as the ...
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19 views

Orbit closures of symmetric bilinear form

Let $A$ and $B$ be two real symmetric matrices in $M_n(\mathbb{R})$. I would like to learn about necessary and sufficient conditions for knowing when $B \in \overline{GL_n(\mathbb{R})\cdot A}$; where: ...
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1answer
21 views

Is there a text introducing “high order Fréchet derivative” well?

Let $X,Y$ be Banach spaces and $U$ be open in $X$. High-order Fréchet derivatives are defined inductively so that the n-th Fréchet-derivative of a function $F$ is $F^{(n)}:U\rightarrow ...
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41 views

Converting tensor product from one coordinate to another

This is a long multi-steps question and I'm stuck at the last leg. I believe my question to be trivial but after 3 hrs of staring and trying all sort of methods (ridiculous ones even) I'm not getting ...
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28 views

Integer such that there is a $k$-algebra isomorphism for any two algebras.

Is there an integer $\ell = \ell(m, n) \ge 1$ such that for any $k$-algebras $A$ and $B$ there is a $k$-algebra isomorphism $\text{M}_m(A) \otimes_k \text{M}_n(B) \cong \text{M}_\ell(A \otimes_k B)$?
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Let $f_1,f_2,f_3 \neq 0$ linear operators in $\mathbb{R}³$, then $\exists y\in\mathbb{R}³,y\neq0 $, {$f_1(y),f_2(y),f_3(y)$} is l.d.

What I did: Let $f_1,f_2,f_3: \mathbb{R}³ \rightarrow \mathbb{R}³$ linear operators and take $\exists y\in\mathbb{R}³,y\neq0 $. Let $\alpha, \beta, \gamma \in \mathbb{R}$ such that $\alpha f_1(y) + ...
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Followup to my previous question, $M = \text{Image}(u^\infty) \oplus \text{Ker}(u^\infty)$.

See my previous question here, Intersection of images and union of kernels. Let $A$ be a ring (not necessarily commutative), let $M$ be an $A$-module, and let $u: M \to M$ be an $A$-module ...
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1answer
36 views

Closed formulas for two Poincaré series

Associated with an arbitrary direct sum $E = \bigoplus_{i \ge 0} E_i$, of finite dimensional $k$-vector spaces $E_i$, $i = 0, 1, 2, \dots,$ there is a formal power series $P_E$, with nonnegative ...
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1answer
79 views

$Alt(T)=0$ if $T$ is a symmetric tensor

Question is to prove that $Alt(T)=0$ if $T$ is a symmetric tensor. We have $$Alt(T)=\sum_{\sigma}sgn(\sigma)T^{\sigma}$$ As $T$ is symmetric we have $T^{\sigma}=T$ for all $\sigma$. So, we have ...
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1answer
61 views

Multilinear Algebra Proof of the Cayley-Hamilton Theorem.

I am trying to understand the proof of the Cayley-Hamilton Theorem given in Section 4 of this document. On pg. 4 of the document, there is a line which reads: From general multilinear algebra, we ...
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1answer
32 views

Can every 2 form be represented as a linear combination of these specific two forms?

This question is Question 2 from Ilka's book on page 8. The first part is to prove that every $\omega^2\in \Lambda^2(V^{\ast})$ can be represented as \begin{equation*}\tag{1} ...
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2answers
103 views

Determinant and alternating multilinear function

Let $V$ be a vector space of dimension $n$ with basis $\{v_1,\cdots,v_n\}$. Let $\phi$ be an n-alternating multilinear map and $A:V\rightarrow V$ is any map (matrix form) then we have to prove that ...
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1answer
48 views

Understanding the algebra of polynomials on a linear space

My advisor and I are working through a paper on partition functions, and we got to the following passage: Fix $n \in \mathbb N$ and let $W := ((\mathbb R^n)^{\otimes 3})^{C_3}$, where the $C_3$ ...
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28 views

Does $\mathfrak T^r(\Bbb R^m)$ count as an vector space?

Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the ...
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Help me to prove the determinant formula

Actually it is about the question of n-linear function, but it is so relevant to the determinant formula. Here is the notation of the theorem. If $n>1$ and $A$ is an $n \times n$ matrix over $K$, ...
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36 views

Wedge product of maps: functorial vs. exterior algebra

Suppose that $V$ and $W$ are finite-dimensional vector spaces over $\mathbb{F}$. If $\varphi, \psi \in \hom(V,W)$, there are at least two interpretations of the symbol $\varphi \wedge \psi$: It is ...
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1answer
33 views

Identification between wedge product and its dual

Let $\mathbb{F}$ be a field, and let $(e_i)$ be the usual elementary basis of $\mathbb{F}^n$. Let $\varphi_{ij}: \mathbb{F}^n \wedge \mathbb{F}^n \to \mathbb{F}$ be such that $v \wedge w \mapsto ...
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Is There a Basis Free Definition of the Pfaffian

$\DeclareMathOperator{\pf}{pf}$ I recently came across a delightful fact that: The determinant of a $2n\times 2n$ skew-symmetric matrix is a the square of a certain polynomial called the pfaffian. I ...
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1answer
70 views

The algebra of natural transformations of the n-th power tensor functor

Let $k$ be a $0$ characteristic field, $n$ an positive integer and $S_n$ the $n$-th symmetric group. Let's work in the symmetric monoidal category of $k$-vector spaces and linear maps that we denote ...
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1answer
32 views

Couple stress tensor reference.

Can someone give me a good mathematical reference for couple stress tensor in its most basic form. Thank you.
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1answer
77 views

Proving Things About Rings Using Things About Vector Spaces

All rings below are assumed to be commutative and having an identity. $\newcommand{\bw}{\bigwedge}\newcommand{\R}{\mathbf R}\newcommand{\mc}{\mathcal}$ Consider the following problem: Problem 1. ...
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1answer
102 views

For $T\in \mathcal L(V)$, we have $\text{adj}(T)T=(\det T)I$.

Let $V$ be an $n$-dimensional vector space over a field of characteristic $0$. For a linear operator $T\in \mathcal L(V)$, we know that $\bigwedge^n T=(\det T)I$, where $I:V\to V$ is the identity map. ...
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18 views

3D Line Integral

I would realy appreciate your kind generous help with this one problem I'm working on in my Engineering Math Intro, Evaluate $ \int_{c} xyz $ ds where C is the curve given by $ \vec ...
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24 views

Natural bilinear map $B\colon Alt^p(E^*)\times Alt^p(E)\rightarrow\mathbb R$

$Alt^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\ \ , u \text{ is alternating multilinear map}\}$ $Alt^P(E):=\{ \alpha\colon ...
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1answer
46 views

Basic Confusion About Tensor Products

Let $A$ and $B$ be subspaces of vector spaces $V$ and $W$ respectively. Given $a\in A$ and $b\in B$, there are two possible interpretations of $a\otimes b$: we can think of it as a member of ...
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The space of alternating multilinear maps and existence of a bilinear map [duplicate]

Before, I ask similar this. But here I change question settings since it was incomplete. I hope receive good ideas. Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as ...
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2answers
60 views

Differential at a point and differential (Differential Geometry)

Given $f\in C^\infty(U)$, $U$ open set of $\mathbb{R}^n$, we define the differential of $f$ at $p$ $$ (df)_p:T_p\mathbb{R}^n\to\mathbb{R}\\ (df)_p(v):=v(f) $$ and the differential of $f$ $$ df:U\to ...
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1answer
29 views

Some calculations with skew forms and wedge product

I have some problems with the language of multilinear forms. I have to prove that if $dim(V)\le 3$, then every $\omega\in\Lambda^q(V^\ast)$ is such that $\omega\wedge\omega=0$. I consider the case ...
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1answer
35 views

Exercise about wedge product and multilinear forms

I'm considering $\omega\in \Lambda^{2q+1}(V^\ast)$, i.e. a multilinear skew-symmetric form. I want to prove that $\omega\wedge\omega=0$. How shall I proceed? Any suggestions? Do I have to write ...
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Isn't this article in wikipedia wrong? (Multilinear form)

https://en.wikipedia.org/wiki/Multilinear_form It says "The exceptional case of characteristic $2$ does not share this property." at last lines. However, isn't it false? Let $R$ be a commutative ...
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1answer
51 views

Question about Grassmannian, most vectors in $\bigwedge^k V$ are not completely decomposable? [closed]

My question: Is $e_1 \wedge e_2 + e_3 \wedge e_4 \in \bigwedge^2 V$ not completely decomposable if $e_1$, $e_2$, $e_3$, $e_4$ is a basis for $V$?
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Basic question about multilinear forms

Let $V$ be a vector space on a field $\mathbb{K}$. Let $(V^\ast)^{\otimes q}$ be the space of $q$-linear forms, $\Lambda^q(V^\ast)$ the space of skew-symmetric multilinear forms and $Sym^q(V^\ast)$ ...
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Basic exercise differential forms

I have to show that the space of q-differential forms $\Omega^q(U)=\{0\}$ if and only if $q>n$ or $q<0$. Any ideas?
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1answer
59 views

Star operator in the simplest form

Let $E$ together with $g$ be a inner product space(over field $\mathbb R$) , $\text{dim}E=n<\infty$ and $\{e_1,\cdots,e_n\}$ is orthonormal basis of $E$ that $\{e^1,\cdots,e^n\}$ is its dual ...
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2answers
49 views

Inner Product on $\Lambda^n(E)$

Let $E$ together with $g$ be a inner product space(over field $\mathbb R$) , $\text{dim}E=n<\infty$ and $\{e_1,\cdots,e_n\}$ is orthonormal basis of $E$ that $\{e^1,\cdots,e^n\}$ is its dual basis. ...
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0answers
46 views

Equivalence between definitions of Tensor Products (Of Vector Spaces)

I've read the definitions in the book of Kostrikin and in the book of Sterling Berberian. The book of Berberian gives a definition for the product of two spacees, Kostrikin gives it more generally. ...
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Are all 7-dimensional cross products isomorphic?

Let $\times$ be this 7-dimensional cross product and let $\hspace{.04 in}f$ be a bilinear map on $\mathbb{R}^7$ which satisfies the orthogonality and magnitude conditions. Does there necessarily ...
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Matrix of $f^p:\Lambda^p(E)\rightarrow \Lambda^p(E)$

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior(alternative) tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
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Pullback maps and an equallity

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
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1answer
90 views

Determinant of the Transpose of an Operator.

Let $V$ be a vector space over a field $F$ of characteristic $0$. A linear operator $T$ on $V$ induces a linear operator $\Lambda^k T:\Lambda^k V\to \Lambda^k V$ such that $\Lambda^k T(v_1\wedge ...
2
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1answer
63 views

Kernel of the Symmetrizing Map $Sym:\bigotimes^k V\to \bigotimes^k V$

$\DeclareMathOperator{\sym}{Sym}$ Let $V$ be a finite dimensional vector space over a field of characterisitc $0$ and $\sym:\bigotimes^k V\to \bigotimes^k V$ be the map given by $$ ...
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Let $A,B:V\to V$ positive definite operators in complex linear space with inner product $V$, $dimV<\infty$

Let $$A,B:V\to V$$ positive definite operators in complex linear space with inner product $$V$$, $$dimV<\infty$$ Show that $$log det(A\cdot B^{-1})=-\int_{0}^\infty tr(e^{-t\cdot A}-e^{-t\cdot ...
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1answer
40 views

Tensor Product of Vectors

let $S,T$ be respectively $k$-, $n$-tensors; $k,n>0$. Then we define the tensor product $$ T \otimes S(x_1,x_2,\ldots,x_{k+n}) := T(x_1,\ldots,x_k) S(x_{k+1}, \ldots, x_{k+n}) $$ (their product as ...
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Definition of Tensors Over Complex Numbers

My question is two part. First, how does the definition of tensors and tensor spaces change when the vectors that the tensors act upon are elements of a complex vector space as apposed to when they ...
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Mahalanobis distance in $\mathbb R^3$ or more dimension

Does anyone know how Mahalanobis distance looks like in $\mathbb R^3$ or $n$ dimensions. I would also be grateful if someone could give a geometrical interpretation in $\mathbb R^3$? Thank you a lot.
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More on the $n$-dimensional cross product: Orientation

Wikipedia states: This formula is identical in structure to the determinant formula for the normal cross product in $\mathbb R^3$ except that the row of basis vectors is the last row in the ...