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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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Can a nonsurjective mapping be a tensor product?

Let $f: R^2 \times R^2 \rightarrow R^{2 \times 2}$ s.t. $ f((x_1, y_1 ),(x_1, y_1 )) = \begin{bmatrix} x_{1} y_1 & x_{1} y_2 \\ x_{2} y_1 & x_{2} y_2 \end{bmatrix} $ f is bilinear, but not surjective. I am told that that (f, $R^{2 \times 2}$)…
fdzsfhaS
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non-commutative multilinear maps?

we introduced multilinear maps today with the following definitions: Let $V$ be a finite-dimensional real vector-space and $k,l\in \mathbb{N}$. A map $T:V\times...\times V\cong V^k\rightarrow \mathbb{R}$ is multilinear, if it is linear in every…
Tobi92sr
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Multilinear map over dimension 1 vector spaces

I'm looking at the trilinear example in this Wikipedia article. I would like to translate this example to the case $f:R\times R\times R\rightarrow R$, where $R$ is the real number line. The basis vector set for each vector space $R^1=R$ is…
Joseph Johnston
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Prove properties of the alternating multilinear map

Let $V$ be a vector space over a field $K$ (with $\operatorname{char}(K) \neq 2$) and let $F:L^n(V,K)\to L^n(V,K)$ be a map defined by $$F(g)(x_1, \dotsc, x_n) := \frac{1}{n!} \sum_{\sigma \in S_n}…
ralf.ra
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How to expand $f(a_1 + h_1, \ldots, a_k + h_k)$?

Suppose $f : \mathbb{R}^{n_1} \times \ldots \times \mathbb{R}^{n_k} \to \mathbb{R}^p$ is multilinear. Suppose $a = (a_1, \ldots, a_k)$ and $h = (h_1, \ldots, h_k)$ are in $\mathbb{R}^{n_1} \times \ldots \times \mathbb{R}^{n_k}$. I would like to know…
Anamaki
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$\wedge$-product is "natural"

I want to proof the following statement: The $\wedge$-product is "natural". Therefor $f^{\ast}\omega\wedge f^\ast\eta=f^\ast(\omega\wedge\eta)$ for every linear function $f:W\to V$ and every $\omega\in Alt^r V, \eta\in Alt^s V$. With the definition…
MathUnicorn
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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 $\ (v_1,\ldots,v_n)\ and\;\ (w_1,\ldots,w_m)$ are bases of V and W respectively, then…
Vishesh
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Complex functions and $\star_3$

I wanted to maybe extend Hodge star/ Technical question to a new question so others could benefit from the idea. So there we discussed that when the $\star$ is Hodge duality star then it is real-linear, by that, if say $\omega$ is a 1-form, then…
PhilosophicalPhysics
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The $n$-fold wedge product of a $2$ form

For the $2$-form $\omega$ on $\Bbb R^{2n}$, $$\omega = \sum_{i = 1}^{2n-1} dx_i \wedge dx_{i + 1}$$ why is $\bigwedge_{i = 1}^n \omega \neq 0$? I thought that if $n = 1$, (testing just $3$ terms) $\omega = dx_1 \wedge dx_2 + dx_3 \wedge dx_4 $. So…
Lemon
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