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

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3
votes
0answers
82 views

Is $M\to M^{\vee\vee}$ injective when $M$ is free?

It's a common theorem that when $M$ is a finite-free $R$-module of rank $n$, there is a natural isomorphism $M\cong M^{\vee\vee}$, where $M^\vee$ denotes the dual. So $M^{\vee\vee}$ is also free of ...
3
votes
1answer
102 views

Why is $M_A\otimes_A N\cong M\otimes_R N$?

I've been doing some tensoring, but am having a hard time understanding the following isomorphism. Suppose $A$ is a commutative $R$-algebra, and for any $R$-module $M$, denote by $M_A=A\otimes_R M$. ...
2
votes
1answer
273 views

Why is the the $k$-th derivative a symmetric multilinear map?

I am having trouble understanding, why the $k$-th derivative of a map $F\colon\mathbb R^n \to\mathbb R^m$ is a symmetric multilinear map for each $x$ in $\mathbb R^n$. Can you please explain which ...
0
votes
1answer
1k views

Hermitian matrices that commute

My question is: If $A$ and $B$ are two Hermitian matrices, and $AB$ is also a Hermitian matrix, then how do prove that both $A$ and $B$ are diagonalizable through the same unitary matrix (i.e the ...
2
votes
1answer
129 views

Symmetric power and characters

Let $V$ be a 2 dimensional vector space over $\mathbb{C}$. Then $W := Sym^{n}(Sym^{m}V)$ is a representation of $GL(V)$. For $g \in GL(V)$, I consider $\chi_{W}(g)$. Let $x$ and $y$ denote the ...
8
votes
2answers
348 views

Tensor Decomposition

Consider a tensor product $$ V^{\otimes n} = \underbrace{V\otimes\cdots\otimes V}_{n} $$ where $V$ is a vector space over $\mathbb R$, $\dim V = m$ , hence $\dim V^{\otimes n} = m^n$ . So every $A ...
1
vote
2answers
320 views

Exterior Power of a Vector Space

Let $G$ be a finite group, $V$ an $n$-dimensional vector space over $\mathbb{C}$, and $\tau: G \rightarrow GL(V)$ a representation such that $\tau(g)$ has determinant 1 for all $g \in G$. Why is it ...
4
votes
1answer
413 views

Orthogonal Complements in Vector Spaces

If $V$ is a finite dimensional vector space over any field $F$, we define an inner product on $V$ as a map $\langle \,, \rangle\colon V\times V\rightarrow F$, satisfying, $\langle u,v+w\rangle ...
1
vote
1answer
245 views

What is the name of these equations?

$xy=0$ $ax +by +cxy +d=0$ $ax +by +cz +dxy +eyz +gxyz=0$ I made myself the examples, sometimes I face these equations and I do not know how to resolve them, all equations whose unknowns have ...
0
votes
2answers
102 views

Is a multilinear form/mapping a product of some type on vectors?

Added: are all types of mappings for vector spaces with "product" in their names always multilinear mappings between some vector spaces? Are there many counterexamples? $F$ is a field. Any ...
1
vote
0answers
53 views

Alternating forms tangential to a subspace.

Let $V$ be a finite-dimensional vector space with euclidean product, and let $U$ be a subspace. Now let $P$ be the projection of $V$ onto $U$, and let $\omega$ be any alternating multilinear $k$-form. ...
5
votes
1answer
847 views

Covectors $\omega^1, …, \omega^k$ are linearly dependent iff their wedge product is zero

How can I prove that covectors $\omega^1, ..., \omega^k$ are linearly independent iff their wedge product $\omega^1\wedge ...\wedge \omega^k$ is not zero?
1
vote
2answers
62 views

Uniqueness of the determinant given some properties

Let be $\varphi:\mathbb C^{2\times 2}\to\mathbb C$ with the following properties: $$$$ It is linear on the columns: $$\left\{\begin{align} ...
0
votes
1answer
77 views

A system of nonlinear equations

Does the following system of six simultaneous equations in eight variables $x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4$ have solutions in $\mathbb{R}$? in $\mathbb{C}$? $$x_1y_2-x_2y_1=1$$ $$x_1y_3-x_3y_1=0$$ ...
1
vote
2answers
286 views

Wedge product and linear subspace

I am trying to understand the relationship between the wedge product and linear subspace. Let $e_1,\cdots, e_4$ be the standard basis of $\mathbb{R}^4$. The wedge product $$(e_1+2e_2)\wedge ...
4
votes
1answer
215 views

On Chevalley's linear identification of the Clifford algebra $C(\mathbf p)$ with the exterior algebra $\wedge \mathbf p$

In reading Sternberg's notes on Clifford algebras and spin representations (page 148) I encountered the following: "...Consider the linear map $$C(\mathbf p)\rightarrow \wedge \mathbf p, x\mapsto ...
1
vote
1answer
234 views

Elementary symmetric polynomials and matrices of 1-forms

Let $A$ be a $n \times n$ matrix of 1-forms (for example, a connection form). Note that $A \wedge A$ is not $0$, but by using the anti-symmetry of the wedge product applied to the entries of $A$ we ...
15
votes
2answers
244 views

Help deriving that $\mathrm{sign} : S_n\to \{\pm 1\}$ is multiplicative

$\def\sign{\operatorname{sign}}$ For homework, I am trying to show that $\sign:S_n \to \{\pm 1\}$ is multiplicative, i.e. that for any permutations $\sigma_1,\sigma_2$ we have $$\sign(\sigma_1 ...
12
votes
1answer
497 views

Polarization: etymology question

The polarization identity expresses a symmetric bilinear form on a vector space in terms of its associated quadratic form: $$ \langle v,w\rangle = \frac{1}{2}(Q(v+w) - Q(v) - Q(w)), $$ where $Q(v) ...
3
votes
2answers
2k views

Basis for tensor products

Suppose $V_1$ and $V_2$ are $k$-vector spaces with bases $(e_{i1})$ and $(e_{i2})$, respectively. I've seen the claim that the collection of elements of the form $e_{i1} \otimes e_{i2}$ forms a basis ...
96
votes
7answers
5k views

Does a “cubic” matrix exist?

Well, I've heard that a "cubic" matrix would exist and I thought: would it be like a magic cube? And more: does it even have a determinant - and other properties? I'm a young student, so... please ...
2
votes
0answers
124 views

Quadratic transformations of vector spaces

Much is known about transformations of the following form $$y_i = L_{ij}x_j \;\;: \;\; x\in\mathcal{R}^n, L\in\mathcal{R}^{n\times n}$$ We can infer a number of geometric properties about the ...
3
votes
1answer
420 views

Universal Definition for Pullback

The concept of "pullback" has several definitions depending on the context in which it is applied, e.g., smooth functions on manifolds, differential forms, multilinear forms and so forth. See, for ...
6
votes
1answer
328 views

Cones of positive semidefinite matrices generated by matrices of rank $1$

Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive ...
0
votes
2answers
229 views

Symmetric Linear Transformations with trivial kernels

Let $V$ be a vector space. Let $A$ be a symmetric bounded multi-linear operator from $V \times V \rightarrow \Bbb{R}$. Suppose that $A(v,v) \neq 0$ for all $v \in V \setminus \{0\}$. This let us ...
10
votes
2answers
1k views

Are “differential forms” an algebraic approach to multivariable calculus?

I am recently learning some basic differential geometry. As I understand, differential forms provide a neat way to deal with the topics in calculus such as Stoke's theorem. In order to define the ...
6
votes
2answers
289 views

Decomposition of product of exterior products

Suppose $V$ is a $n$-dimensional vector space. What is the kernel of $$\bigwedge^p V \otimes \bigwedge^q V\longrightarrow \bigwedge^{p+q} V$$ here $p+q \le n$.
1
vote
1answer
196 views

Canonical Isomorphism Between $\Omega^2(\mathbb{R^3})$ and $\mathbb{R^3}$?

Let $\Omega^2(\mathbb{R}^3)$ represent the collection of differential 2-forms on $\mathbb{R}^3$. For this space we take as an (ordered) basis $\{dx \wedge dy, dx \wedge dz, dy \wedge dz\}$. First ...
3
votes
2answers
211 views

Bases for exterior powers

I've seen the following claim several times: If $V$ is a vector space over $K$ with basis $\{e_1,\ldots,e_n\}$ then the basis to the kth exterior power of $V$ is given by the elements ...
2
votes
1answer
353 views

Dimension of vector space of all $n$-linear functions

Let $F$ be a field. I am trying to compute the dimension of the vector space of all $n$ linear functions $D:F^k \rightarrow F$ I figured this is a pretty standard calculation but the only place so ...
3
votes
1answer
349 views

What does “a map is isomorphic to another map” mean?

In this article, there are two proposition as the following: 1.(Proposition.) For every bilinear map $f:V\times W\to U$ there is a unique linear map $h:V\otimes W\to U$ such that $hg=f$, where $g$ ...
4
votes
2answers
157 views

How should I understand “every bilinear function $f$ on $V\times W$”?

In this article, there is a lemma as following: Let $U$ and $V$ be vector spaces, and let $b:U\times V\to X$ be a bilinear map from $U\times V$ to a vector space $X$. Suppose that for every ...
15
votes
4answers
4k views

understanding of the “tensor product of vector spaces”

In Gowers's article "How to lose your fear of tensor products", he uses two ways to construct the tensor product of two vector spaces $V$ and $W$. The following are the two ways I understand: ...
8
votes
1answer
296 views

Questions about bilinear maps from a Gowers's article

The following is from Gowers's How to lose your fear of tensor products: Because subscripts and superscripts are a nuisance in html, I shall now change notation, and imagine that we know the ...
3
votes
1answer
504 views

understanding of the tensor product $V^*\otimes W^*$

In S.S.Chern's Lectures on Differential Geometry, I don't understand the following text in Chapter 2, which introduces the tensor product: The tensor product $V^*\otimes W^*$ of the vector spaces ...
6
votes
5answers
2k views

Concrete Example Illustrating the Interior Product

Let $V$ be a finite-dimensional vector space, let $v \in V$ and let $\omega$ be an alternating $k$-tensor on $V$, i.e., $\omega \in \Lambda^{k}(V)$. Then, the interior product of $v$ with $w$, denoted ...
9
votes
2answers
364 views

Subspace intersecting many other subspaces

V is a vector space of dimension 7. There are 5 subspaces of dimension four. I want to find a two dimensional subspace such that it intersects at least once with all the 5 subspaces. Edit: All the 5 ...
4
votes
1answer
214 views

Do any non-combinatorial proofs of the elementary properties of wedge products exist?

The wege product, an operation defined between two alternating tensors, has a number of elementary properties such as associativity, distributivity, etc. There are many proofs of these properties ...
11
votes
3answers
588 views

Signs in the natural map $\Lambda^k V \otimes \Lambda^k V^* \to \Bbbk$

Let $V$ be a finite-dimensional vector space over a field $\Bbbk$. Let $V^*$ denote its dual. I strongly suspect that there is a natural map $$\Lambda^k V \otimes \Lambda^k V^* \to \Bbbk$$ that ...
4
votes
2answers
1k views

Mathematically Precise Definition of Covariant and Contravariant Transformation

I am trying to understand the meanings of "covariant transformation" and "contravariant transformation" and how they are related. I have read the related Wikipedia article and still feel I cannot ...
8
votes
3answers
350 views

Two introductory linear algebra problems

I remember when I was in Moscow one of my homework questions was: Is there a $2\times 4$ matrix whose $2\times 2$ minors are: a) $(2,3,4,5,6,7)$ b) $(3,4,5,6,7,8)$ c) ...
5
votes
1answer
232 views

Extension of Riemannian Metric to Higher Forms

I've been reading about Riemannian manifolds, and have come across a comment that says that for a metric $g$ on an $N$-dimensional manifold $M$, considered as a bilinear map $$ g:\Omega^1(M) \times ...