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

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2
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0answers
27 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 ...
2
votes
0answers
62 views

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 $K$, ...
0
votes
0answers
34 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 ...
1
vote
1answer
29 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 ...
5
votes
0answers
39 views

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 ...
4
votes
1answer
66 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 ...
2
votes
1answer
28 views

Couple stress tensor reference.

Can someone give me a good mathematical reference for couple stress tensor in its most basic form. Thank you.
7
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1answer
73 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. ...
6
votes
1answer
95 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. ...
0
votes
0answers
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 ...
2
votes
0answers
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 ...
3
votes
1answer
45 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 ...
1
vote
0answers
35 views

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 ...
4
votes
2answers
58 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 ...
0
votes
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 ...
1
vote
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 ...
4
votes
0answers
54 views

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 ...
0
votes
1answer
49 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 $V$?
3
votes
0answers
55 views

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 $Sym^q(V^\ast)$ ...
0
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0answers
28 views

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?
3
votes
1answer
58 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 ...
5
votes
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. ...
2
votes
0answers
42 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. ...
3
votes
0answers
70 views

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 ...
1
vote
0answers
45 views

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 ...
2
votes
0answers
35 views

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 ...
5
votes
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
votes
1answer
62 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 $$ ...
1
vote
0answers
21 views

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 ...
0
votes
1answer
38 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 ...
0
votes
0answers
29 views

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 ...
2
votes
0answers
17 views

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.
0
votes
0answers
22 views

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 ...
1
vote
2answers
27 views

What is the right hand side in this definition of $n$-dimensional cross product

Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basis for $\mathbb{R}^n$ and let ${\bf w_1} = (w_{11},\dots,w_{1n}), \dots, {\bf w_{n-1}}=(w_{n-1\;1},\dots,w_{n-1\;n}) \in \mathbb{R}^n$. Then one ...
0
votes
2answers
28 views

Tensor manipulation

I am very new at manipulating tensors and I have the following equation: $$A_{\mu \nu\tau} b^\mu c^\nu = g_{\tau \rho} d^\rho$$ where $\tau$ is the independent index and $g_{\tau \rho}$ the metric ...
1
vote
0answers
25 views

Do tensor norms exist?

Does there exist norms for tensors, as an extension for the ordinary matrix norm? For example, if there is a derivative of a matrix [A] with respect to a vector {x}, does the norm of this derivative ...
2
votes
0answers
38 views

Tensor transpose notation

I have a rank 3 tensor $\mathbf{Q}$. What notation should I use to denote the transposition of two of the dimensions? For instance, if I want to transpose the first and second dimensions, one way I ...
4
votes
2answers
61 views

Linear Maps as Tensors

Let $V$ and $W$ be finite dimensional vector spaces and let $V^{\ast}$ denote the dual $V$. I read that the space $V^{\ast}\otimes W$ may be thought of in four different ways: as the space of linear ...
3
votes
1answer
25 views

Tensor, exterior, symmetric powers over fields of nonzero characteristic

I was reading Fulton and Harris' discussion of exterior and symmetric products as quotient spaces of tensor products in their rep theory book when I noticed that they made this claim (the emphasis is ...
0
votes
0answers
24 views

What's the name of this tensor product?

Fix $V$ to be a vector space over $\mathbb{R}$. For all $k \in \mathbb{N}$, let $L_k$ be the space of all $k$-tensors on $V$, and let $S_k$ be the set of all permutations of the set $\{1,\dots, k\}$. ...
2
votes
1answer
63 views

Endomorphism commutes with its adjugate

Let $R$ be a commutative ring, $M$ a free $R$-module of rank $n$ and $f \in \rm{End}(M)$. The adjugate $f^\sharp$ of $f$ is defined by the equalities $$ f^\sharp(x) \wedge y = x \wedge ...
1
vote
1answer
34 views

Find the projection of a vector onto a subspace of $\Bbb R^4$

I need to find the projection of $\vec b = (1,1,1,1)$ onto a subspace of $\Bbb R^4$ described as: $$V=\{(x,y,z,t)\,:\,x=y+t\ \hbox{and}\ 2x=y+z\}\ .$$ Thanks for any help i get guys.
1
vote
1answer
46 views

If a Bilinear Form is Non-Degenerate on a Subspace $W$, then $V=W\oplus W^\perp$.

$\newcommand{\range}{\text{image}}\newcommand{\ann}{\text{Ann}}\newcommand{\set}[1]{\{#1\}}$ Problem: Let $V$ be a finite dimensional vector space over a field $F$ and $f$ be a symmetric bilinear ...
0
votes
1answer
36 views

Representing a series of Matrix inner product with a single matrix product.

I have a set of constraints in my optimization problem, constraints in the form , $\langle A, e_i e_j^T \rangle = r_{ij} ,\forall i,j \epsilon S$, where $A$ is an $n*n$ semidefinite and symmetric ...
4
votes
1answer
86 views

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$ This is not homework, it is part of an answer of Show that $\mathbb{A}_\mathbb{C}^2 \ncong ...
1
vote
0answers
17 views

The space $V^{0}_{p}$ of p times covariant tensors and canonical isomorphisms

I have been studying tensor calculus by myself, but I have found the following claim in my book: The space $V^{0}_{p}=V^{*} \otimes \cdots \otimes V^{*}$ of $p$ times covariant tensors is ...
-2
votes
1answer
28 views

Have a question about Linear Transformations

Explain why there cannot be a linear transformation T: $R^2$ --> $R^2$ for which T(1,1)=(2,3) and T(3,3)=(1,4). I have no clue how to start this problem. Wouldn't ...
4
votes
0answers
55 views

When does the duality functor commute with the wedge power functor?

When working with modules over a fixed commutative ring, I know that $(M \otimes N)^* \cong M^* \otimes N^*$ provided either $M$ or $N$ is finitely generated projective. Does it follow that ...
-1
votes
1answer
26 views

What is the best fit (in the sense of least-squares) to the data?

A) Find the best fit (in the sense of least-squares) to the data $x_1$ $(1,-1,-1,1)$ $x_2$ $(1,1,-1,-1)$ $y$ $(5,1,1,1)$ by a linear function of the form $y$=$a$+$bx_1$+$cx_2$ B) Find ...
8
votes
2answers
38 views

$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?