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

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Proving an equivalence relation $(A_1,B_1)$ ~ $(A_2,B_2)$

Let $(A_1,B_1),(A_2,B_2) \in \Bbb R^{n\times n} \times \Bbb R^{n \times m}$. We say that $(A_1,B_1)$ and $(A_2,B_2)$ are similar written $(A_1,B_1)$ ~ $(A_2,B_2)$, if there exists $S \in$ GL (n, $\Bbb ...
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52 views

CW complex such that action induces action of group ring on cellular chain complex.

Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ on the space $\overline{X}$ given ...
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30 views

Associativity of the exterior multiplication of forms.

Let $\omega^{k}\in\bigwedge^{k}(V^{*})$ and $\eta^{l}\in\bigwedge^{l}(V^{*})$, be two exterior forms of degrees $k$ and $l$. The exterior product $\omega^{k}\wedge\eta^{l}$ is defined as $(k+l)$ form, ...
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2answers
59 views

Bilinear maps and functions of the form $(x,y) \mapsto ux^2 + 2vxy + wy^2$

I was recently reading from the book "Vectors, Pure and Applied", $\S 16.1$ on bilinear forms. It begins In section 8.3 we discussed functions of the form $$(x,y) \mapsto ux^2 + 2vxy + wy^2$$ ...
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34 views

How to show that the isotropic tensor of order n is a multiple of the kronecker delta

I have already found this question here but with the property of invariant under rotation. However I don't have this property and I want to prove that $T_{ij} = \alpha \delta _{ij}$ where $T_{ij}$ ...
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1answer
73 views

Is $J(V)$ just the symmetric algebra of the dual of $V$?

Let us work over a fixed but arbitrary field $k$. So for example, all vectorspaces and algebras are implicitly over $k$. Given a vectorspace $V$, we can build two (potentially) different commutative ...
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58 views

Tensor product of $k$-algebras, center, isomorphism.

Let $A$, $B$ be two $k$-algebras of finite dimension, where $k$ is a field. Here, $A$ and $B$ are not necessarily commutative. Do we have that$$Z(A \otimes_k B) \cong Z(A) \otimes_k Z(B),$$where ...
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1answer
59 views

Does the tensor product with a finite dimensional vector space gives you some kind of semi-basis?

Let $V$ be a (possibly infinite dimensional) vector space, and $W$ a finite dimensional one. Then, one can define the tensor product $V\otimes W$ as the free vector space on $V\times W$ modulo some ...
3
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1answer
54 views

Over the reals the intersection of the orthogonal and symplectic groups in even dimension is isomorphic to the unitary group in the half dimension.

See the answer here. Over the reals the intersection of the orthogonal and symplectic groups in even dimension is isomorphic to the unitary group in the half dimension:$$ U(n) = O(2n, ...
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1answer
35 views

The $-1$-eigenspace of the exchange map is $V \wedge V$

Let $V$ be a vector space, and $T(\mathbf{v}_0 \otimes \mathbf{v}_1) = \mathbf{v}_1 \otimes \mathbf{v}_0: V \otimes V \to V \otimes V$ be a linear map. Theorem: The eigenspace of $T$ with ...
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2answers
29 views

Show R is a basis for Poly ; Find coordinate change matrix

Given $Poly=\{ a_1 t^2 + a_2t + a_3\}$ ; and $B=\{t^2,t,1\}$ is a basis $Poly$. a) Show that $R=\{t^2 +1 , t-2 , t+3\}$ is a basis for $Poly$ b) Find the coordinate change matrix $S$ from the basis ...
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1answer
43 views

Basis of a Tensor Product

I was wondering if anyone could explain why the following proof of linear independence is not valid. Choose the identity function, $i: V \times W \to V \times W$. This function is clearly bilinear so ...
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1answer
58 views

Linear maps preserving the determinant and Hermiticity

Conjecture: Let $H_n$ be the space of $n\times n$ complex Hermitian matrices and let $\varphi:H_n \to H_n$ be a linear map which preserves determinants: \begin{equation} \det \circ \varphi = \det. ...
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1answer
27 views

How the canonical symplectic form acts

I've read that the canonical symplectic form $\omega$ on $\mathbb R^{2n}$ is given by $$\omega=\sum_{i=1}^n dp_i\wedge dq_i,$$ where $(p_1,\dots,p_n,q_1,\dots,q_n)$ are the coordinates on $\mathbb ...
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0answers
49 views

Do the structure constants of $\mathfrak{su}(n)$ specify an injective map?

The structure constants of a Lie algebra are determined by $[T^a, T^b] = f^{ab}\,_c T^c$. As a $(2,1)$ tensor, they can however be considered as a linear map from vectors to antisymmetric tensors. Is ...
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208 views

Finding the area of a parallelogram on a 3D coordinate plane

I'm stuck on the following question; I've attempted it but I'm not sure if I've done it right. Find the area of the parallelogram on the plane (−x + 3y + 3z = 6) defined by 0 ≤ x ≤ 3, 0 ≤ y ≤ 2. So ...
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44 views

Compression of a matrix A by V

I can't understand and even can't find any text on Compression of a matrix A by V. meaning if $B=V^*AV$ then B is called the compression of A. What does it mean???
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1answer
46 views

How to create an equation with 3 variables, given two points in 3-dimensions?

I have two points: (20.33, 16.1, 0.0150) & (20.48, 19, 0.0123), and I would like an equation of the line that connects these two points, but I'm not entirely sure how to do it. Is there a form of ...
2
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1answer
69 views

Exterior derivative commutes with postcomposition by symmetric multilinear functionals?

Let $\frak{g}$ be a finite-dimensional real Lie algebra, $\varphi: \bigotimes^l \frak{g} \to \mathbb{R}$ a symmetric multilinear functional, and $\psi \in \Omega^k(M; \bigotimes^l \frak{g})$ a ...
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0answers
24 views

Does the tensor product operation distribute over the (external) direct sum?

It has been asked before if it holds that $B\otimes (C\oplus D)\cong (B\otimes C)\oplus (B\otimes D)$, for B,C,D all being real vector spaces. Is it known if the analogous statement, $(B\otimes ...
2
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1answer
66 views

Kronecker delta and Levi-Civita symbol

Prove that $$ det \left[ \begin{matrix} δ_{ak} & δ_{al} & δ_{am} \\ δ_{bk} & δ_{bl} & δ_{bm} \\ δ_{ck} & δ_{cl} & δ_{cm} \\ ...
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1answer
225 views

What is the “star product” of vectors really called, and where can I learn more about it?

Background conventions: Identify each natural number with the corresponding von Neumann ordinal. e.g. $$3 = \{2,1,0\}$$ Think of $\mathbb{R}^n$ as the set of all functions $n \rightarrow ...
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1answer
75 views

Curiosity about the wedge product and the Levi-Civita symbol

Let us use the definition of the wedge product of two vectors: $$\vec{u}\wedge\vec{v} = \vec{u}\otimes\vec{v} - \vec{v}\otimes\vec{u}$$ writing $\vec{u}$ and $\vec{v}$ in dyadic form as $\vec{u} = ...
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50 views

Approach to build multivariate ranking system?

I have data of various sellers on ecommerce platform. I am trying to compute seller ranking score based on various features, such as 1] Order fulfillment rates [numeric] 2] Order cancel rate ...
6
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1answer
43 views

Exterior 2-form, 1-form, Hodge star operator.

In $\mathbb{R}^{2n}$ with coordinates $x_1, x_2, \dots, x_{2n}$, consider an exterior 2-form$$\eta = \sum_{k=1}^n x_{2k-1} \wedge x_{2k}.$$Given a 1-form $\alpha = \sum_{i=1}^{2n} a_ix_i$, what is the ...
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35 views

Multilinear extension of submodular function

I am reading the wikipedia article about submodular functions. Let $\Omega$ be a finite set and $f\colon 2^\Omega\to \Bbb R$ a submodular set function, i.e. a function such that $$f(S)+f(T)\geq ...
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Proof of Simple Properties of Volume

Let $e_{1},\ldots,e_{n}$ be vectors in $\mathbb{R}^{n}$. Define a parallelepiped $P$ to be a translate of the set $$\left\{x\in\mathbb{R}^{n} : x=t^{1}e_{1}+\cdots+t^{n}e_{n}, 0\leq t^{i}\leq ...
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1answer
64 views

Inference of an identity in Grassmann algebra.

I am reading Herbert Federer's book called "Geometric Measure Theory", in chapter one of Grassmann algebra, on pages 36-37, he says that for $f$ being an endomorphism of a finite dimensional inner ...
4
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1answer
118 views

What do we call the result of wedging together the columns of a matrix?

We can wedge together the columns of a square matrix to compute its determinant. More generally, the exterior product of the columns of a $b \times a$ matrix tells us the determinant of each $a \times ...
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1answer
68 views

Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = ...
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1answer
36 views

Orbit closures of real symmetric bilinear form

Let $\alpha$ and $\beta$ be two real symmetric bilinear forms in $\operatorname{sym}(\mathbb{R}^n)$, with signatures $(p_{\alpha},n_{\alpha},z_{\alpha})$ and $(p_{\beta},n_{\beta},z_{\beta})$. I ...
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1answer
33 views

Alternating multilinear function satisfies $f(A)=\det(A)f(Id)$

I've just seen a proof of the statement: "Given $\alpha$ in a commutative ring $K$ there is a unique alternating multilinear function $f$ with $f(Id)=\alpha$." The determinant is defined as the ...
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Orbit closures of symmetric bilinear form

Let $A$ and $B$ be two real symmetric matrices in $M_n(\mathbb{R})$. I would like to learn about necessary and sufficient conditions for knowing when $B \in \overline{GL_n(\mathbb{R})\cdot A}$; where: ...
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1answer
29 views

Is there a text introducing “high order Fréchet derivative” well?

Let $X,Y$ be Banach spaces and $U$ be open in $X$. High-order Fréchet derivatives are defined inductively so that the n-th Fréchet-derivative of a function $F$ is $F^{(n)}:U\rightarrow ...
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1answer
54 views

Converting tensor product from one coordinate to another

This is a long multi-steps question and I'm stuck at the last leg. I believe my question to be trivial but after 3 hrs of staring and trying all sort of methods (ridiculous ones even) I'm not getting ...
4
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34 views

Integer such that there is a $k$-algebra isomorphism for any two algebras.

Is there an integer $\ell = \ell(m, n) \ge 1$ such that for any $k$-algebras $A$ and $B$ there is a $k$-algebra isomorphism $\text{M}_m(A) \otimes_k \text{M}_n(B) \cong \text{M}_\ell(A \otimes_k B)$?
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1answer
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Let $f_1,f_2,f_3 \neq 0$ linear operators in $\mathbb{R}³$, then $\exists y\in\mathbb{R}³,y\neq0 $, {$f_1(y),f_2(y),f_3(y)$} is l.d.

What I did: Let $f_1,f_2,f_3: \mathbb{R}³ \rightarrow \mathbb{R}³$ linear operators and take $\exists y\in\mathbb{R}³,y\neq0 $. Let $\alpha, \beta, \gamma \in \mathbb{R}$ such that $\alpha f_1(y) + ...
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1answer
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Followup to my previous question, $M = \text{Image}(u^\infty) \oplus \text{Ker}(u^\infty)$.

See my previous question here, Intersection of images and union of kernels. Let $A$ be a ring (not necessarily commutative), let $M$ be an $A$-module, and let $u: M \to M$ be an $A$-module ...
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1answer
38 views

Closed formulas for two Poincaré series

Associated with an arbitrary direct sum $E = \bigoplus_{i \ge 0} E_i$, of finite dimensional $k$-vector spaces $E_i$, $i = 0, 1, 2, \dots,$ there is a formal power series $P_E$, with nonnegative ...
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1answer
86 views

$Alt(T)=0$ if $T$ is a symmetric tensor

Question is to prove that $Alt(T)=0$ if $T$ is a symmetric tensor. We have $$Alt(T)=\sum_{\sigma}sgn(\sigma)T^{\sigma}$$ As $T$ is symmetric we have $T^{\sigma}=T$ for all $\sigma$. So, we have ...
2
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1answer
70 views

Multilinear Algebra Proof of the Cayley-Hamilton Theorem.

I am trying to understand the proof of the Cayley-Hamilton Theorem given in Section 4 of this document. On pg. 4 of the document, there is a line which reads: From general multilinear algebra, we ...
2
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1answer
38 views

Can every 2 form be represented as a linear combination of these specific two forms?

This question is Question 2 from Ilka's book on page 8. The first part is to prove that every $\omega^2\in \Lambda^2(V^{\ast})$ can be represented as \begin{equation*}\tag{1} ...
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2answers
228 views

Determinant and alternating multilinear function

Let $V$ be a vector space of dimension $n$ with basis $\{v_1,\cdots,v_n\}$. Let $\phi$ be an n-alternating multilinear map and $A:V\rightarrow V$ is any map (matrix form) then we have to prove that ...
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1answer
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Understanding the algebra of polynomials on a linear space

My advisor and I are working through a paper on partition functions, and we got to the following passage: Fix $n \in \mathbb N$ and let $W := ((\mathbb R^n)^{\otimes 3})^{C_3}$, where the $C_3$ ...
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0answers
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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 ...
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106 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 $K$, ...
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47 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 ...
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2answers
62 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
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0answers
62 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
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1answer
81 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 ...