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

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0
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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
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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 ...
7
votes
1answer
74 views

Direct proof of non-flatness

Consider $k$ a field and the rings $A=k[X^2,X^3]\subset B=k[X]$. How to prove that $B$ is not flat over $A$ by using only the definition of flatness that it maintains exact sequences after making ...
1
vote
1answer
80 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
66 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
55 views

Tensor Product definition (help with certain step)

I'm going over some notes I took from the blackboard, and reached a slight hitch. I thought that maybe someone could help. Let $E,F$ be vector spaces over a field $\mathbb K$. A tensor product of $E$ ...
2
votes
1answer
44 views

Notation in Bleecker's Gauge Theory and Variational Principles

In the proof of the theorem that there is a unique linear isomorphism $\star:\bigwedge^k(E)\to\bigwedge^{n-k}$ on p.4 in Bleecker's Gauge Theory and Variational Principles he says For ...
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 $\mathcal{Cl}(V,q)$?
0
votes
1answer
79 views

Tensor product and valence of a tensor?

Given a vector space $V$, its dual vector space $V^{*}$ and a tensor $\mathbf{T}$: $\mathbf{T} \in \underbrace{V \otimes\dots\otimes V}_{n\text{ copies}}\otimes \underbrace{V^{*}\otimes\dots\otimes ...
1
vote
0answers
81 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 $1$. ...
4
votes
1answer
104 views

Tensor basis change

I have a question regarding tensors and basis changes, however upon searching the web I've found an infinity of definitions for tensors so I'll have to give the one I know first: Given a vector space ...
1
vote
1answer
216 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
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0answers
44 views

Second derivative dot product

I was wondering whether I am correct that the second derivative of the dot product is this: Let $f:X \times X \rightarrow \mathbb{K}$ $(x_1,x_2) \mapsto \langle x_1,x_2 \rangle$ Then we have ...
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
1answer
28 views

sufficient condition for being an integral factor

Let $ f: \mathbb {R}^m \rightarrow \mathbb {R}-\{0\} $ function $C^{\infty}$ class and $w$ a one-form $C^{\infty}$ class in $\mathbb {R}^m $. If $\alpha=w-\dfrac{1}{f}dx_{m+1} $ satisfies $\alpha ...
1
vote
1answer
37 views

Dimension of (p,q) forms

Let $E$ be a complex vector space of dimension $n$. What is the dimension of the multilinear alternate forms on $E$ of type $(p,q)$ ? I'm sure this is classical but I couldn't find a reference, and ...
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, ...
0
votes
0answers
44 views

Tensor power of a quotient space

Let $E$ and $F$ be vector spaces, $F$ a subspace of $E$. Is there any canonical isomorphism between $E/F \otimes E/F$ and a quotient of the form $E \otimes E/G$, where $G$ is a subspace of $E \otimes ...
4
votes
1answer
103 views

Is invariance of a multi-linear form required for co/contra variance?

I'm reading the book: The Absolute Differential Calculus by Levi-Civita to get an idea of the history behind the development of tensor calculus. On page 71 he states: An m-fold covariant is an ...
2
votes
1answer
53 views

Tensor Product and Direct Sum

Let $R$ be a commutative ring with identity and let $\{M_\alpha\}$ be a family of $R$-modules and $N$ another $R$-module. I've tried to show that $$\left(\bigoplus_\alpha M_\alpha\right)\otimes ...
2
votes
0answers
93 views

Recognizing pure tensors in tensor product of vector spaces

Let $V$ be a vector space and let $\{e_i\}$ be a basis for it. Then $\{e_I\equiv e_{i_1}\otimes...\otimes e_{i_r}\}$ is a basis for $V\otimes ... \otimes V$. Suppose I am given an element $w=\sum a_I ...
0
votes
1answer
29 views

Is this proof that $\mathcal{L}(V_1,\dots,V_k;W)\simeq \mathcal{L}(V_1;\mathcal{L}(V_2,\dots,V_k;W))$ correct?

I've been trying to show that if $V_1,\dots,V_k,W$ are vector spaces over $K$, then $$\mathcal{L}(V_1,\dots,V_k;W)\simeq \mathcal{L}(V_1;\mathcal{L}(V_2,\dots,V_k;W)),$$ I think I've got the idea, ...
1
vote
1answer
153 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
1answer
66 views

Existence of isomorphism between tensor products.

In multilinear algebra many maps are usually proven to exist rather than simply defined. For example, commutativity is one such example. In the book I'm studying the author says: let $V_1,\dots,V_k$ ...
6
votes
3answers
248 views

Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
4
votes
2answers
135 views

Tensor Multiplication - Why should we use permutations?

I'm reading a book on multilinear algebra, and the author first establishes this easy isomorphism: if $V_1,\dots,V_k$ are vector spaces over the field $K$ and if $\sigma\in S_k$, then there is an ...
2
votes
3answers
112 views

How to really understand the tensor algebra?

If $V$ is a vector space over $F$, then we define $T^r_0(V)=V^{\otimes r}$, then we define the algebra of contravariant tensors to be $$T(V)=\bigoplus_{r=0}^\infty T^r_0(V)$$ together with the ...
1
vote
1answer
52 views

Why it makes sense to think of multivectors as “paralelograms”?

Let $V$ be a vector space over the field $\mathbb{K}$ and let $T(V)$ be it's tensor algebra. We usually define the exterior algebra $\Lambda (V)$ by the following process: we consider the bilateral ...
1
vote
2answers
106 views

Why in differential geometry tensors are usually defined as multilinear maps?

In multilinear algebra books tensors are usually defined through the universal property. Given a family of $k$ vector spaces $V_1,\dots,V_k$ over the same field $F$ we want to construct a space $S$ ...
3
votes
1answer
122 views

Poincare duality and Hodge duality

The Poicare duality is defined in Greub's Multilinear algebra (1967) in Chapter 6, §2 as a isomorphism between $\bigwedge V$ and $\bigwedge V^*$, where $V$ is a finite-dimensional vector space, $V^*$ ...
1
vote
1answer
158 views

Tensors = matrices + covariance/contravariance?

I have read several topics on tensors but it is still not clear to me. Tensors are different from matrices because they contain additional information about how do they transform. To fully specify a ...
3
votes
4answers
139 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 ...
2
votes
2answers
107 views

p-forms as multilinear maps

I'm studying differential geometry and am learning about differential forms. We have a very intuitive and simple way to understand 1-forms as linear maps on from the tangent space to the base field, ...
2
votes
2answers
126 views

Eisenbud's proof of right-exactness of the exterior algebra

I'm trying to understand the proof in Eisenbud's Commutative Algebra that, given a right exact sequence $$K \to N \to M \to 0$$ of $R$-modules, we have an exact sequence $$K \otimes \wedge N \to ...
0
votes
0answers
25 views

Result depending on a lot of variable where a change won't really matter much, any principle to describe this?

I have a value $A =f(p_1,p_2,p_3,...,p_N)<0$ with certain values of $\bar p$. The function $f$ is a multilinear function. Now the value of $A$ is very rarely above $0$ with a lot of different ...
0
votes
0answers
188 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha ...
4
votes
1answer
227 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 ...
7
votes
0answers
226 views

Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
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
80 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
1answer
104 views

Lipschitz condition in infinite dimensional vector spaces

If we have that $T:V \times W \rightarrow Y$ multilinear and $V,W$ are infinite-dimensional normed vector spaces.(the finite-dimensional proof is easy, since you can use compactness of the boundary ...
1
vote
0answers
55 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 ...
3
votes
2answers
62 views

A clarification regarding partial derivatives

Let us suppose the $i^{th}$ partial derivative of $f:\Bbb{R}^n\to \Bbb{R}$ exists at $P$; i.e. if $P=(x_1,x_2,\dots,x^n)$, $$\frac{f(x_1,x_2,\dots,x_n+\Delta x_n)-f(x_1,x_2,\dots,x_n)}{\Delta ...
1
vote
1answer
40 views

Terminology for a generalization of multilinearity to any algebraic structure.

Let $S$ and $T$ be sets with the same algebraic structure. Let $\Phi:S^n\to T$ such that for any $i\leq n$, and $s_1,\ldots,s_n\in S$, the aplication $s\in S\mapsto ...
1
vote
0answers
72 views

Methods to minimise multilinear functions with trilinear, quad-linear and higher-linear terms?

My goal is to minimize functions such as $$f_1(\mathbf{p})=p_1p_3p_7+p_1p_4p_7+p_2p_3p_7+p_2p_4p_7-p_1p_3p_5p_6-p_1p_4p_5p_6-p_2p_3p_5p_6-p_2p_4p_5p_6$$ and ...