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

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Decomposition into simple bivectors

According to Wikipedia, any element of $\wedge^2\Bbb R^n$ should be decomposable into $n/2$ simple bivectors for $n$ even or $(n-1)/2$ for $n$ odd. How do I count that? How do I check that ...
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29 views

2 forms and Base

Let$\: V \;$ be a n-dimensional vector space and $\:w\;$ a two form. Proof that there exists a base $\alpha_1,\alpha_2,..\alpha_n, \in V^* \;$ so that $\; \omega =\alpha_1 \wedge \alpha_2 + \alpha_2 ...
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75 views

Book recommendation for rigorous multilinear algebra , tensor analysis, manifolds.

I am looking for recommendation on books about multilinear algebra, tensor analysis, manifolds theory, basically everything to be able to understand basic concepts of general relativity. I am ...
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1answer
41 views

Confusion regarding notation of a dual transformation

I'm reading Spivak's Calculus on Manifolds and in Chapter 4 he defines the dual transformation (although he doesn't call it that) as follows: If $f:V \rightarrow W$ is a linear transformation, a ...
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1answer
53 views

Generalization of scalar product for vectors in n-dimensional space

Let $x$ and $y$ be two vectors and $A$ the angle between them. Then we have the scalar product $$x\cdot y = \|x\|\|y\| \cos A$$ Let $x$, $y$ and $z$ be three vectors; $A$ angle between $x$ and $y$; ...
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43 views

Dual space of exterior power and exterior power of dual space

Let $V$ be a finite-dimensional vector space. Is there an isomorphism between $\Lambda^k(V^\ast)$ and $\left(\Lambda^k(V)\right)^\ast$? I was able to prove this with the additional requirement of ...
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1answer
60 views

Map to exterior power gives rise to smooth embedding of Grassmannian in projective space?

How do I see that the map$$(x_1, \dots, x_n) \mapsto x_1 \wedge \dots \wedge x_n$$from $V_n(\mathbb{R}^m)$ to the exterior power $\wedge^n(\mathbb{R}^m)$ gives rise to a smooth embedding of ...
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33 views

Prove that every polynomial from a finite Banach space to another Banach space has an unique representation in terms of coordinate functionals

I'm trying to solve Problem 1.2.J in Mujica's "Complex Analysis in Banach Spaces". The problem states as follows: Let $E$ and $F$ be Banach spaces over $\mathbb{K}$, with $E$ finite dimensional. ...
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19 views

Block-matrix constructions for outer product of sets of vectors and higher order tensors.

So I'm in the process of doing a large sequence of vector-vector outer-products. On my software it would be convenient to do this in a systematic vectorizable way. Say I have my $n$-dimensional column ...
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58 views

Meaning of / intuition for contraction of tensors (in the Riemannian setting)

I'm currently taking a course in differential geometry, and we are, I'm guessing, finally going to start working with the Riemannian curvature tensor after having covered a lot of smooth manifold ...
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1answer
109 views

A coordinate free book on linear and multilinear algebra defining determinants using exterior algebra

I would like to find an advanced introduction to linear and multilinear algebra that is 1)Coordinate free 2)Use tensor products and exterior algebras to define determinants 3)DOES NOT assume a ...
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1answer
74 views

On volume forms and norms on exterior powers

Let $V$ be a $2$-dimensional vector space. Given an inner product on $V$ one may define an inner product on the simple $k$-vectors of $\Lambda^k(V)$ by $$\langle v_1 \wedge \cdots \wedge v_k, w_1 ...
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1answer
44 views

How can I prove that the norm of the following operator is $\frac{1}{m!}$?

I'm trying to prove that the norm of the multilinear symmetric operator $A$ is $\frac{1}{m!}$ where $A$ is defined as: $$ A(x_1,\dots, x_m) = \frac{1}{m!} \sum_{\sigma \in S_m} \xi_1(x_{\sigma(1)} ) ...
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37 views

Erdos-Ko-Rado theorem when k divides n

So i'm trying to prove Erdos-Ko-Rado for a Kneser graph $K(k,n)$ when I know that $k$ divides $n$. If $C$ is some coclique of $K(k,n)$ I've argued that for any ordered partition $(A_1,..,A_{n/k})$ of ...
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1answer
53 views

Integral of exponential with linear term

$$x \in \mathbb{R^n},$$ M is a positive symmetrical nonsingular nxn Matrix and j is an arbitrary vector in $$\mathbb{R}^n.$$ The following has to be calculated: $$Z(j) = \int_\mathbb{R^n} ...
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1answer
29 views

cardinality of orbits

Let $G=\mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p )\times \mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p ) \times \mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p )$, $p$ prime, act on $(\mathbb{Z}/p\times ...
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22 views

Concrete description of altenating or symmetric power functor (analogous to Kronecker product)

Suppose that an $A$-module homomorphism is given between two free and finite dimensional $A$ modules $M$ and $N$, with basis $e_1 \ldots e_n $ and $f_1 \ldots f_m$ respectively, by a matrix $T \in ...
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1answer
37 views

Understanding a step in a computation involving dual basis and permutations

Since $\varphi_{i_j} \in \mathcal{T}(\mathbb{R}^n)$, for every $j = 1, \dots, k$, we have \begin{align*} \varphi_{i_1}\wedge\dots\wedge\varphi_{i_k}(e_{i_1}, \dots, e_{i_k}) &= ...
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1answer
172 views

Proving that $\phi$ is an alternating multilinear map

Let $V$ be a finite-dimensional vector space over the field $F$ with basis $\mathcal B = \{v_1,\dots,v_n\}$. Let $1\leq k\leq n$ and pick some $1\leq i_1 < i_2 < \dots < i_k \leq n$. I am ...
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1answer
26 views

Manipulation of skew-symmetric linear map

Let $\Delta$ be a skew-symmetric $n$-linear map. I have the following in my notes and I am having trouble seeing how it follows: $$ \Delta\left(\sum_{i=1}^n{e_i}, \sum_{i=1}^n{(e_i)} -e_2, ...
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1answer
47 views

Differential and Rank of $XAX^{-1}-A$

I have a map: $F_{A} (X) :GL\left(2n,\mathbb{R}\right) \longrightarrow\mathbb{\mathfrak{M}_{\mathit{2n\times2n}}\left(\mathbb{R}\right)}$ such as \begin{eqnarray} & F_{A}(X) & =XAX^{-1}-A ...
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22 views

Trace form seen by tensorial product.

I'm studying tensor product by the book "Multilinear Algebra - W. H. Greub". On page $38$ I couldn't understand how he got into the equation $(1.41)$. Could anyone help me?
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1answer
66 views

Generalized “scalar product” based on multilinear form?

In an $\mathbb{R}$-vector space $V$, the scalar product is a paradigmatic example of a non-degenerate, symmetric, positive-definite bilinear form $\beta : V \times V \to \mathbb{R}$. I wonder if the ...
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37 views

Programming nested sums in Matlab for graph-based statistic

I have an undirected graph $G=(E,N)$, where $E$ is the set of edges and $N$ is the set of nodes, of which $|N|=n$. It's convenient to represent the edges via a (symmetric) adjacency matrix $B$. I ...
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1answer
56 views

$M$ is finitely generated as an $A$-module iff $M/A_{>0}M$ is finitely generated as an $A$-module?

Let $A$ be a nonnegative graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$. How do I see that $M$ is finitely generated as an $A$-module if ...
3
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1answer
61 views

Signature of the inner product.

I need some help with this problem. Consider in $\mathbb{R}^{4}$ the lorentz's metric $h$ which has a signature $(3,1)$, this mean that diagonal matrix $M(h)$ is the form $$M(h)=\begin{pmatrix} 1 ...
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1answer
24 views

derivative of kronecker product

Given $x \in \mathbb{R}^N$ and a function $$H = \sum_{i,j,k=1}^n\ J_{i,j,k}\ x_i x_j x_k$$ for a fixed $J \in \mathbb{R}^{n \times n \times n}$, I am trying to calculate the derivative $\frac{d H}{d ...
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Grammatically confused: $\omega=4dV$ for 3-form $\omega$ and volume in $\Bbb R^4$?

Background: Against the advice I should have been given but wasn't, I'm taking a Lie theory course with no background in differential geometry. We finally made it into the part of the course where we ...
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28 views

Composition of linear maps as a tensor product

$\mathbf{Question:}$ Let $V$ be a finite-dimensional vector space over an algebraically closed field $F$. Fix $A,B \in \mathscr{L}(V)$. Consider the linear operator $T_{A,B} \in ...
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1answer
49 views

How to understand acting one tensor on another tensor to obtain a third tensor?

I've already known the definition of the tensor that a tensor T of type $(k,l)$ is a multilinear map from a collection of dual vectors and vectors to $\mathbf{R}$: $$ T: ...
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1answer
30 views

Dimension of the basis $\{x \otimes y + y \otimes x\}$

I'm trying to prove that the annihilator of $I = \left<x \otimes y - y \otimes x \right>$ is $\left<x \otimes y + y \otimes x \right>$. To do this I am trying to compare dimensions. So if ...
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30 views

Problem with triple cross product proof

When trying to prove bac-cab rule, I get to point, where I don´t know, what is true. I have $$ (\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})a_jb_lc_m=a_jb_ic_j-a_lb_lc_i $$ but when $$ ...
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1answer
52 views

raising/ lowering indices

Here is my understanding of tensors: There is more than one way to think about tensors. One way is be thinking about tensors as objects with components which obey some transformation laws. For ...
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1answer
22 views

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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1answer
60 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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32 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
74 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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1answer
39 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
75 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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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
66 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
61 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
39 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
35 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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54 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
67 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
30 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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54 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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223 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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45 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???