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Questions tagged [multilinear-algebra]

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

A multilinear function is a function from $V_1\times V_2\times\dots \times V_n\to V$ where the $V_i$ and $V$ are all vector spaces, and the function is a linear function into $V$ when restricted to each of the $V_i$. This includes the special case of bilinear functions.

For example, the ordinary dot product in $\Bbb R^n=V$ is a multilinear function from $V\times V\to \Bbb R$.

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$\wedge$-product, alternating

I want to show, that the $\wedge$-product of multilinear forms is alternating. Therefore $\omega\wedge\eta(\dotso, v,\dotso, v,\dotso)=0$ Since $\omega$ and $\eta$ are already alternating multilinear forms, there is only one interesting case. The…
MathUnicorn
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dual basis, $\wedge$-product

I have to following statement, which I want to proof. Unfortunatly I am not used to the dual space and know barely anything about it. I want to proof what seems trivial. Namely: Let $e_1,\dotso, e_k$ be the canonical basis of $\mathbb{R}^k$ and let…
MathUnicorn
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Wedge product is non-degenerate

Let $V$ be a k-dimensional vector space and $m,n \in \mathbb{Z}$ are such that $m+n \le k$. We consider the wedge product: $\wedge: \Lambda^nV \times \Lambda^mV \to \Lambda^{m+n}V$ defined by $(v,w) \to \pi(v \otimes w)$ where $\pi$ is…
Wooster
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Step in Proof of Existence of Hodge-*

I am following the proof of the existence of the Hodge-* operator in Naber's Geometry, Topology and Guage Fields. Given a basis $(e_1, \dots, e_n)$ for a vector space and it dual $(e^1, \dots, e^n)$ suppose that $$ \sum_{i_1 < \cdots < i_k}…
ItsNotObvious
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About non-degenerate skew symmetric form

The definition of a non-degenerate skew symmetric $\omega : H \otimes V \to H^{*} \otimes V^{*} $, where H and V are finite dimensional vector spaces, is that for each $v \in V$ non-zero, $\omega : H \otimes \langle v \rangle \to H^{*} \otimes V^{*}…
User43029
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Multilinear maps: is $\phi(av_1,v_2)$ always equal to $\phi(v_1,av_2)$?

I am learning about multilinear maps by myself and the book I'm following gives a definition which is somewhat vague. That's the definition: Given vector spaces $V_1,V_2,\dots,V_p,W$. A mapping $\phi:V_1\times\dots\times V_p \rightarrow W$ is…
Thiago
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Induced Action of Matrix on Tensor Product

I've been asked to write the induced action of a matrix in $M_4(\mathbb{R})$ on $\mathbb{R}^4 \otimes \mathbb R^4$, but this terminology is unfamiliar to me. What does it mean for a matrix to induce an action on a tensor product? Similarly, what…
Empiromancer
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Question about determinant as transformation on alternating multilinear $n$-forms

If $T:E \to F$ is a linear transformation, $f : F\times\cdots\times F \rightarrow \mathbb{R}$ is an alternating, multilinear $n$-form and $\overline{T}:A_n(F)\rightarrow A_n(E)$ is a function that assigns an alternating, multilinear $n$-form in $E$…
Gui
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Symmetric tensors and duals

Let V be a finite dimensional vector space and consider $(Sym^n V^\vee)^\vee$ where $\vee$ denotes thedual, i.e homogenous polynomials in V of degree n. Consider as well $S_n(V)$, consisting of fixed points of the canonical action of $\Sigma_n$ ,…
Marvelaction
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Classification of Multilinear Functions

In An Introduction to Manifolds, Tu defines: "Denote $V^{k} = V \times \dots \times V$ the Cartesian product of k copies of a real vector space V. A function $f: V^{k} \mapsto \mathbb{R}$ is k-linear if it is linear in each of its k arguments." As…
Scott Sobolewski
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Multilinear functions of degree zero

I apologize in advance, but I need help with a rather simple question. Given $L_k(V)$ is the set of all $k-$linear functions on the vector space $V$ on $\mathbb{R}$, what is $L_0(V)$? Does it make sense? And why? I cannot simply get there from the…
Giuseppe
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Exercise from Greub, Multilinear Algebra on factorization of bilinear maps

I am really stuck on exercise 5 in chapter 1.3 of Greub's book Multilinear Algebra. The exercise asks me to show that if $\varphi \colon E \times F \to G$ and $\psi \colon E \times F \to H$ are bilinear maps such that $N_1(\varphi) \subseteq…
user920957
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$C(S \times T)$ is isomorphic to $C(S) \otimes C(T).$

Let $S$ and $T$ be two arbitrary sets and consider the vector spaces $C(S)$ and $C(T)$ generated respectively by S and T. Show that $C(S \times T)$ is isomorphic to $C(S) \otimes C(T).$ I am starting to read Werner Greub's Multilinear Algebra and I…
wessi
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n-form is zero implies linear dependence.

Let $\omega\in \Lambda^nV$ be an $n$-form for some real linear space $V$. It is known that for $v_1, \dots,v_n\in V$: $$\omega(v_1,\dots,v_n)=0\iff \{v_1, \dots,v_n\} \ \text{is linearly dependent.}$$ The $\Leftarrow$ implication is easily done. I…
Stijn D'hondt
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Question about exterior product and tensor product

I'm reading about the element of volume: let $(V,\varphi)$ be an oriented Euclidean vector space; then exists a unique alternating tensor $\sigma\in\mathcal A^d(V)$ s.t. $\sigma(v_1,\dots,v_d)=1$, when $(v_1,\dots,v_d)$ is a positively oriented and…
Dr. Scotti
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