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

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3
votes
0answers
170 views

How to invert this function on matrices which involves the permanent?

I'm interested in understanding whether a particular natural function on matrices, closely related to the permanent of a matrix, is invertible, and whether its inverse admits a simple closed form. The ...
1
vote
1answer
37 views

Can $W(V)\hookrightarrow\operatorname{End}_k(R)$ when $\operatorname{char}(k)=0$?

This is a follow up about a case I'd been meaning to ask in a question I asked about a week ago. Suppose $V$ is a vector space of dimension $2n$, and let $W(V)$ be the associated Weyl algebra, which ...
0
votes
2answers
248 views

Levi-Civita Symbol and index manipulation

Above is the question and answer to question (b) (ignore (a))... I don't get where the final implication comes from. Why can we use c as a (covariant) index on the LHS... surely we must use d or ...
5
votes
1answer
393 views

An Expression for the Wedge Product

For the question below, I have the following definitions and concepts in mind: The $k^{th}$ exterior power of a real vector space $V$, denoted $\Lambda^k(V)$ can be realized as the quotient of the ...
13
votes
2answers
833 views

Proving that the coefficients of the characteristic polynomial are the traces of the exterior powers

Let $T$ be an endomorphism of a finite-dimensional vector space $V$. Let $$f(x)=x^n+c_1x^{n-1}+ \dots + c_n$$ be the characteristic polynomial of $T$. It is well known that ...
7
votes
2answers
303 views

Why is the following map an isomorphism between $Cl(V,\omega)$ and $\operatorname{End}(\Lambda(V))$?

Suppose you have a vector space $V$ of dimension $2n$. I know that there exists a basis $x_1,\dots,x_n,y_1,\dots,y_m$ such that $\omega(x_i,x_j)=\omega(y_i,y_j)=0$ and $\omega(x_i,y_j)=\delta_{ij}$, ...
0
votes
2answers
137 views

Question regarding Tensor product

Let $\ T: V\times W \rightarrow \mathbb V\otimes W$ be a map defined as $\ T(v,w) = v\otimes\ w$ where $\ v \in V,w\in W $. Then T is bilinear. Further if $\ (v_1,\ldots,v_n)\ and\;\ ...
3
votes
1answer
99 views

Homomorphism $W(V)\to\operatorname{End}_k(R)$ is not injective when $\operatorname{char}(k)=p>0$?

Suppose $V$ is a vector space of dimension $2n$, and let $W(V)$ be the associated Weyl algebra, which can be viewed as an associative $k$-algebra with generators $x_1,\dots,x_n,y_1,\dots,y_n$ ...
1
vote
1answer
199 views

Exterior Algebra of Self-Direct Sum

Suppose $V$ and $W$ are vector spaces and $\bigwedge V$ and $\bigwedge W$ their exterior algebras. Then it is known that $\bigwedge (V \oplus W) \simeq \bigwedge V \otimes \bigwedge W$. Now my ...
3
votes
1answer
135 views

Subset of differential operators is a finitely generated module?

I was reading about differential operators, and there is a small claim I don't understand. First, let $A$ be a commutative algebra over $k$, a field. We have the recursive definition for the algebra ...
2
votes
2answers
156 views

Exercise at the Beginning of Part II in Fulton's Book on Young Tableaux

In Fulton's Book Young Tableaux, there's an Exercise at the beginning of part II for which I cannot find a solution (there doesn't seem to be one for this exercise in my copy of the book). It reads: ...
8
votes
1answer
812 views

Condition for a tensor to be decomposable

Let $V$ be a vector space of dimension 3 with basis $e_1,e_2,e_3$. Let $W$ be a vector space of dimension 2 with basis $f_1,f_2$. Is $e_1\otimes f_1+e_2\otimes f_2$ decomposable? What about ...
2
votes
1answer
258 views

Basis of the symmetric algebra $S(M)$ given $R$-module basis of $M$ using the diamond lemma?

Over the past week, I read this secret blogging seminar post concerning the diamond lemma, which got me to reading about Bergman's paper on the diamond lemma. Now suppose you have a free $A$-module ...
1
vote
1answer
153 views

A system of three nonlinear equations

I have a system of nonlinear equations. Here it is: $$ \frac{s_2 - K_2}{ps_2^{\gamma_1} + (1 - p)s_2^{\gamma_2}} = \frac{K_1 - s_1}{ps_1^{\gamma_1} + (1 - p)s_1^{\gamma_2}} \\ \frac{s_2}{s_2 - K_2} ...
3
votes
1answer
189 views

Tensors as linear combinations of pure tensors.

Let $V$ be an n-dimensional real vector space, consider the space $F(V^p)$ of real functions on the p-fold cartesian product $V^p$ and its subspace $(V^{*})^p$ of multilinear functions (i.e. covariant ...
2
votes
2answers
93 views

How can one see that $\operatorname{tr}(f\otimes g)=\operatorname{tr}f\operatorname{ tr }g$?

Suppose you have two free modules $M$ and $N$ of finite rank over a commutative ring $R$. Let's also take some $f\in\operatorname{End}_R(M)$ and $g\in\operatorname{End}_R(N)$, which gives a ...
11
votes
2answers
266 views

Why is it that $\det(\phi-x\text{id})=\sum_{i=0}^n (-1)^ic_ix^i$?

I'm trying to understand a certain formula for the determinant in a more general setting. Say you have a free module $M$ of rank $n$ over a (commutative) ring $R$. Let ...
11
votes
1answer
533 views

Sub-determinants of an orthogonal matrix

Let $A$ be a matrix in the special orthogonal group, $A \in SO_n$. This means that $A$ is real, $n \times n$, $A^t A = I$ and $Det(A)=1$, that is, the column vectors of $A$ make a positively-oriented ...
3
votes
0answers
83 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
104 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
297 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
355 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
330 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
418 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
246 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
887 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
63 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
78 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
292 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
216 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
512 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 ...
99
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
128 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
434 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
338 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
238 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
292 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
198 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
226 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
384 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
351 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
158 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 ...