2
votes
0answers
18 views

Exploiting structure in multilinear equations

I'm wondering if there are any standard techniques for exploiting structure in multilinear equations. An example of what I have in mind is solving $A_{ab} X_{bc} A_{cd} (B_{ad} B_{bc} + B_{ac} ...
0
votes
0answers
41 views

How do you solve a linear transformation with no transformation matrix given?

I am stuck, I can't see how Tff was found with no transformation matrix. And now am being asked to find Tgg, help me http://oi60.tinypic.com/33yrplv.jpg
3
votes
1answer
56 views

Is the matrix form of the cross product related to bilinear forms.

The cross product of two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^3$ can be represented as a matrix product as follows, if $\mathbf{x} = (x_1, x_2, x_3)^{\top}$ then $\mathbf{x} \times ...
0
votes
1answer
36 views

Simplying linear equation to get quartic in q using Maple and then using Descarte’s rule of sign

Using the maple I am trying to get quardic in q from this big linear equation. Then use Descarte’s rule of signs to determine the number of positive roots. \begin{equation} ...
2
votes
0answers
28 views

Is there a definition of the Pfaffian parallel to the definition of the determinant in terms of the exterior algebra?

If $V$ is a finite-dimensional vector space, we have some $A \in \operatorname{End}(V)$ then we can extend $A$ to act on the exterior algebra $\bigwedge(A)$ by setting $$A \cdot (v_1 \wedge \cdots ...
0
votes
0answers
64 views

Linear algebra puzzle

I stumbled upon an interesting problem in my calculations, which I can't figure out how to solve directly. It goes as follows: show that $$ -2\left(I\otimes (Z-Q)C^{-1} + (Z-Q)C^{-1} \otimes I ...
2
votes
4answers
213 views

Tensor product in multilinear algebra

In the book by Halmos ($FDVS$) the tensor product of two vector spaces U and V is defined as the dual of the vector space of all the bilinear forms on the direct sum of U and V. Is there a generalised ...
1
vote
1answer
52 views

Trace of the exterior power as a determinant

Let $A$ be a matrix. According to Wikipedia, $$tr(\wedge^k A) = \frac{1}{k!} \det \begin{pmatrix} tr (A) & k-1 & 0 & \cdots \\ tr (A^2) & tr (A) & k-2 & \cdots \\ \cdots & ...
0
votes
1answer
43 views

Ideal of an integral domain all of whose exterior powers are nonzero.

I want to find an integral domain $R$ with ideal $I$ (considered as an $R$-module) such that $\bigwedge^k I\neq 0$ for all nonnegative integers $k$. Dummit and Foote gave the example of $R=\mathbb ...
1
vote
0answers
39 views

The canonical perspective on the Hodge star operator

I am looking for the canonical perspective on the Hodge star operator. I want to see it done properly, not using basis for its definition, saying clearly what we assume in its definition. ...
1
vote
1answer
31 views

Unclear Construction of Basis for Tensor Product

My problem lies in page 363 of Steven Roman's Advanced Linear Algebra (Here's a link). The author says that for each ordered pair $(e_i,f_j)$ where $\left\{e_i\right\}_{i\in ...
0
votes
0answers
19 views
2
votes
1answer
66 views

Is the wedge product surjective?

Is the wedge product $\wedge : \Lambda^{p}(V) \otimes \Lambda^{q}(V) \to \Lambda^{p+q}(V)$ surjective, for $V$ a real vector space of finite dimension? What dimension does its kernel have?
1
vote
1answer
29 views

How to show $\nu=dx_1\wedge\ldots \wedge dx_n$?

Let $\nu$ be the $n$-form in $\mathbb R^n$ satisfying $\nu(e_1, \ldots, e_n)=1$ where $\{e_1, \ldots, e_n\}$ is the canonical base of $\mathbb R^n$. Let $\displaystyle v_i=\sum_{j=1}^n a_{ij}e_i$. How ...
1
vote
0answers
55 views

How to show the differential form $\nu$ satisfies $\nu(v_1, \ldots, v_n)=\det(a_{ij})$?

In $\mathbb R^n$ consider the differential form $\nu$ satisfying $\nu(e_1, \ldots, e_n)=1$. For every $i=1, \ldots, n$ consider the vector $\displaystyle v_i=\sum_{j=1}^n a_{ij} e_j$. How to show ...
2
votes
2answers
59 views

Why there is this relation between $k$-vectors and $k$-forms?

I've been trying to understand the geometrical meaning of $k$-vectors and $k$-forms on some vector space $V$ of finite dimension $n$ over a field $\Bbb K$. Indeed, as I understood, a $k$-form $\omega ...
0
votes
1answer
59 views

linear independent or dependent set - linear algebra

I have the following set: $\{ [1; -1; -2], [-1;0;1], [1;2;1] \}$ and I need to find out whether the set is independent or dependent. My answer and the book's answer contradict. I thought it was ...
0
votes
1answer
56 views

Exterior power and alternating forms: explicit computations

I would like to get a more concrete understanding of a general isomorphism I have read about. I apologize if this is too basic, but I was not satisfied with the references at my disposal. Let $K$ be ...
0
votes
4answers
68 views

$\{u_{i} \otimes w_{j} \}_{i , j}$ forms a basis for $U \otimes W$

Suppose $U$ and $W$ are $k$-vector spaces with bases $\{u_{i}\}_{i=1}^{n}$ and $\{w_{j}\}_{j=1}^{m}$. How to prove that $\{u_{i} \otimes w_{j} \}_{i , j}$ forms a basis for $U \otimes W$ ?
0
votes
1answer
141 views

Linear Transformation induced by the following matrix A

Suppose $T:\mathbb R^4\rightarrow\mathbb R^4$ is the transformation induced by the following matrix $A$. Determine whether $T$ is one-to-one and/or onto. If it is not one-to-one, show this by ...
1
vote
1answer
81 views

Tensor Einstein summation notation

I have two tensors $A^i$ and $B_j$ with components $(2,3,4)$ and $(1,2,3)$ respectively. What is the difference between $A^i B_i$ and $A^i B_j$? Is it just: $A^i B_i = 2+6+12 = 20$ $A^i B_j =$ $ ...
6
votes
0answers
67 views

Is $(V_1\otimes\cdots\otimes V_k)^\ast \simeq V_1^\ast\otimes \cdots \otimes V_k^\ast$ true for infinite dimensional spaces?

Suppose $V_1,\dots,V_k$ are vector spaces of finite dimension. Then I could prove easily that $(V_1\otimes\cdots\otimes V_k)^\ast\simeq V_1^\ast\otimes\cdots\otimes V_k^\ast$. My proof was like that: ...
1
vote
0answers
42 views

Tensor vector bundle construction

$\newcommand{\p}{\partial}$Let $M$ be a smooth manifold, and define $$T_{r,s} := \bigsqcup_{p \in M} (T_p M)_{r,s} = \bigsqcup_{p \in M} \big( \underbrace{T_p M \otimes \dots \otimes T_p M}_r \otimes ...
1
vote
1answer
68 views

How to show that $v_1\otimes\cdots\otimes v_k = 0$ if and only if at least one $v_i = 0$?

I'm trying to show that given vector spaces $V_1,\dots,V_k$ (not necessarily finite dimensional) over the same field $F$ then if $v_i\in V_i$ we have $v_1\otimes\cdots\otimes v_k = 0$ if and only if ...
0
votes
1answer
48 views

$b:V \times W \to \mathbb{R}$ bilinear. Show an induced $\phi:V \to W^*$ surjective.

Let $V$ and $W$ be finite dimensional vector spaces. Let $b:V \times W \to \mathbb{R}$ a bilinear map satisfying: $\forall v \in V. (\forall w \in W. b(v,w)=0) \implies v=0$, $\forall w \in W. ...
2
votes
1answer
49 views

Number of Involutive Automorphisms on a Clifford Algebra

Let $V$ be a vector space with dimension $n$ and $q$ a quadratic form on $V$. How many involutive automorphisms are there in $\mathcal{Cl}(V,q)$?
1
vote
0answers
82 views

Sum of the squares of the minors of a matrix with orthonormal column vectors = 1?

Let $A$ be an $m \times n$ ($n \leq m$) matrix with real entries and orthonormal column vectors. Claim: For $1 < k \leq n$, the sum of the squares of the $k\times k$ minors of $A$ is always $1$. ...
1
vote
1answer
218 views

Orthogonal Complements property

I have a question about how to prove a certain property of orthogonal complements of vector subspaces. Given $\mathrm E$, a vector space over a commutative field k, define: $$\phi\text{ : } \mathrm E ...
0
votes
1answer
84 views

Finding the determinant of an $n\times n$ matrix… and the inverse

Finding the determinant of a $2\times 2$ matrix is easy and the inverse is even easier. Finding the determinant of a $3\times 3$ matrix and its inverse is a little more difficult but still doable. ...
3
votes
1answer
81 views

Understanding higher order SVD

Can someone explain the singular value decomposition of a tensor (maybe a 3 dimensional matrix) with an example? It is intuitively difficult to the get the meaning from just the formulas. On a ...
2
votes
2answers
119 views

Einstein Notation for product of stacked matrices

Background Information: I recently started using the Einstein summation notation to express certain operations over an "image" $\mathbf{A}$ where to each pixel a square matrix is attached. That is, ...
1
vote
1answer
158 views

Direct sum and tensor product of two representations of a group

Our lecturer gave us a hard exercice to go further in group theory (we stopped at group actions) : Let G be a group, V and W complex vector spaces and $\rho_1 : G \mapsto GL(V) $ be a group ...
3
votes
4answers
143 views

What's a good reference to study multilinear algebra?

This semester I'm taking a course in linear algebra and now at the end of the course we came to study the tensor product and multilinear algebra in general. I've already studied this theme in the past ...
4
votes
1answer
228 views

What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity), then the tensor product $f_1\otimes ...
0
votes
2answers
20 views

if $v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is member of $K$. How is this possible?

Let $(V,K)$ and $u,v$ is a member of $V$. Suppose that $M$ is a subset of $V$ is a subspace of $V$ with basis $B_m=\{m_1,...,m_r\}$ with $r$ less than and equal to $n$. Let $H$ be a subspace spanned ...
0
votes
1answer
81 views

Pairing of vector spaces

Suppose we have a pairing i.e. a bilinear map $\phi : V \times V \rightarrow \mathbb{R} $. There are many ways of getting a map $ \psi: V \rightarrow V^* $ two of which are $v\rightarrow \langle ...
1
vote
0answers
56 views

Describing multilinear maps as linear operators

Let a finite dimensional complex vector space $V$ be given. Let $T^k(V)$ denote the vector space of multilinear maps $V^k\to\mathbb C$. My original question was going to be as follows: Does there ...
3
votes
1answer
84 views

Finding all alternating bilinear $T$ that preserve a certain group of isometries of $\mathbb{R}^{n+1}$

Let $$G=\left\{\begin{pmatrix} H & 0 \\ 0 & 1\end{pmatrix} \ | \ H\in O(n), HJ=JH \right\}\subset \mathrm{Lin}(\mathbb{R}^{n+1},\mathbb{R}^{n+1}) $$ where: $n=2m$, $J$ is the standard complex ...
0
votes
1answer
48 views

Determinant of symmetric matrix of the form $v\otimes v$

Note that for $V=\mathbf{R}^n$, $$S^2V = \{ v\otimes w \mid v, w\in V\text{ and }v\otimes w=w\otimes v \} =\{ A\in \mathrm{M}_2(\mathbf{R}) \mid A=A^T \}.$$ Clearly, $S^2V $ contains $O=\{ v\otimes v ...
2
votes
2answers
81 views

Universal property definition from Greub's Multilinear Algebra

Hi I started studying Greub's multilinear algebra book and I found something very strange when he defines the tensor product of two vector spaces: He defines: [...] Let $E$ and $F$ be vector spaces ...
1
vote
0answers
26 views

Terminology for multilinear functionals the sum of whose cyclic shifts is zero?

Let $\varphi$ be an $n$-linear functional on a vector space $X$. Suppose that $\varphi$ has the property that, for all $(x_1,\ldots,x_n) \in X^n$, we have $$ \varphi(x_1,\ldots,x_n) + ...
3
votes
1answer
149 views

Trace of the $n$-th symmetric power of a linear map

Suppose $V$ is a vector space over $k$ and $\dim(V) = N$. Let $A \in\operatorname{End}(V)$. Let $\wedge^n A \in \operatorname{End}(\wedge^n V)$ where $\wedge^n$ is the $n$-th exterior power. I am ...
0
votes
1answer
37 views

Bilinear Form - Proof

I have to prove that the mapping $f(x,y) = {\displaystyle \sum_{i=1} ^ {n} }{ \displaystyle \sum_{j=1}^{n} }x_iy_j{f}(e_i,e_j)$ is a bilinear form, that is, inter alia, the condition: ...
1
vote
2answers
72 views

bilinear form - proof

I have to prove that the mapping $f(x,y)={\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{j=1}^{n}}x_{i}y_{j}{f}(e_{i},e_{j})$ is a bilinear form, that is, inter alia, the condition: ...
7
votes
1answer
282 views

Quick question: tensor product and dual of vector space

Recall that for a finite dimensional vector space $V$ we have the natural isomorphism $\phi :V^{*} \otimes V \rightarrow Hom(V,V)$ given by $\alpha \otimes v \mapsto (x \mapsto \alpha (x)v)$. Is ...
2
votes
0answers
105 views

Computing distances between hyperspheres and sides of a hypercube?

Suppose you are given the $n$ dimensional hypersphere: $$\left(x_1 - \frac{1}{2}\right)^2 + \left(x_2 - \frac{1}{2}\right)^2 +\ldots+ \left(x_n - \frac{1}{2}\right)^2 = \frac{n}{4}$$ And the ...
3
votes
0answers
81 views

Prove that $\phi_1 \wedge \cdots \wedge \phi_k (v_1, \cdots, v_k) = \frac{1}{k!}\det[\phi_i(v_j)].$

I have proved these two exercises: (1) Suppose that $T \in \Lambda^p(V^*)$ and $v_1, \ldots, v_p \in V$ are linearly dependent. Prove that $T(v_1, \ldots, v_p) = 0$ for all $T \in \Lambda^p(V^*)$. ...
13
votes
4answers
324 views

Why the words “inner” and “outer” to designate products?

Does anyone know what's the rationale for using the adjectives inner and outer for certain algebraic products? Also, I've seen the term exterior algebra. Does the exterior here have anything to do ...
2
votes
1answer
139 views

basis-independent isomorphism

Could some one give me some help with this proof? Given the hint, I still don't have a clue about how to proceed. Thanks.
0
votes
0answers
63 views

Multi-affine function

Suppose i have a three-variable function f(x1,x2,x3), f:R^3 -> R. If it is linear for x1,x2 and x3 we can say it has the form f(x1,x2,x3) = c1x1 + c2x2 + c3x3 where c1,c2,c3 in R. We can ...