# Questions tagged [multilinear-algebra]

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

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### Why does a space vector $V$ must have finite dimension to be isomorphic to its bidual $V^{**}$?

There are many different ways to define tensors. Actually it seems that the word "tensor" is applicable to many various concepts/objects. In any case, it also seems that when we use the multilinear ...
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### Intuition behind existence of mixed volumes?

Consider "Volume" as a function from set of $d$-dimensional convex bodies to real numbers. This function is homogeneous of degree $d$ (under rescalings of the convex bodies). Minkowski's theorem ...
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### If $V=k[x]$ then $V \otimes_k V \simeq k[x,y]$

Let $V=k[x]$. Show that $V \otimes_k V \simeq k[x,y]$. I consider the function $\phi : V \otimes_k V \to k[x,y]$ given by $\phi(f(x) \otimes g(y)) = f(x)g(y)$. I could show that $\phi$ is injective, ...
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### How to write an undergraduate level paper on topics of multilinear algebra or tensor?

I'm a undergraduate student majoring in Mathematics. My professor only suggested me considering the topics in multilinear algebra or tensor, but without any other specific instruction. This topic is ...
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### Why are exterior products so much “wigglier” than symmetric and tensor products?

Apologies in advance that this is a somewhat soft question. Let $k$ be an infinite field. Fix a dimension $d$ and let $v_1,\dots,v_r$, $w_1,\dots,w_r$ be two tuples of linearly independent vectors in ...
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### What is the geometric interpretation of a multilinear subspace? [closed]

If we have two linearly independent vectors in n-dimensions, we know that they span a plane, for example. In general, they form a subspace. I got introduced to multilinear algebra somehow, but I ...
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### Mechanics of a $(0,2)$-tensor and a bi-vector acting on 2 vectors

Both bivectors and $(0,2)$-tensors are mathematical structures that take in $2$ vectors and produce a scalar. Similar as in this prior post I wrote, I would like to dumb down the mechanics of these ...
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### Skew-symmetric implies alternating for $2$ a zero divisor in $R$?

In Keith Conrad's notes: https://kconrad.math.uconn.edu/blurbs/linmultialg/extmod.pdf Theorem 2.10 reads: Let $k\geq 2$. If $2\in R^\times$, then a multilinear function $f:M^k \to N$ which is ...
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### Representing a multi-affine map as a determinant

Given a multi-affine map $f: \mathbb{R}^n \mapsto \mathbb{R}$, is it always possible to represent the function as a determinant? And is there a principled way to generate the matrix if it is possible? ...
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### Which subspaces of exterior power have decomposable bases?

Let $V$ be a real $n$-dimensional vector space, and let $1<k<n,r>1$. I wonder: Is there a way to characterise which $r$-dimensional subspaces of the exterior power $\bigwedge^k V$ have ...
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### $G$ is Lie group and $V$ is a representations of $G$,prove representations $V \otimes V \cong S^2(V) \oplus \Lambda^2(V)$

Let $G$ a Lie group and let $V$ a representations of $G$. Then we have the following representations are isomorphic: \begin{align} V \otimes V \cong S^2(V) \oplus \Lambda^2(V) \end{align} I have no ...
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### Is every basis for $\bigwedge^kV$ satisfying a “complementary” property a rescaling of a “standard” basis?

This question was inspired by this beautiful answer: Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$ ($1 \le i_1 < \ldots < i_2 \le 4$) be a basis for $\bigwedge^2V$, ...
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### How to determine an integral of a differential form?

Let $$\eta = x^2 \, dy \wedge dz + yx\,dz \wedge dx + z^3 \, dx \wedge dy$$ Can you show me how to calculate : $$\int_{\Phi} \eta,$$ where $\Phi$ is supposed to be the parametrization of ...
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### Relationship between anisotropic and negative/positive definite

Let $K$ be an ordered field, and $(V, <->)$ a non-degen. bilinear space, where $<->$ is a bilinear form. Determine whether the following statement is true or false: $(V,<->)$ is ...
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### Determining values of multilinear forms/ differential forms

I started getting into the topic of multilinear forms and differential forms. I find it quite hard to get into. if I solve  \Phi ( \begin{pmatrix} 1 \\ 2\\3 \end{pmatrix} , \begin{pmatrix} 4 \\5\\6 ...
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### Convert from one tensor canonical form to another

Suppose we have two canonical forms $A, B \in \mathbb{F}_2^{2 \times 2 \times 2}$ of a 3-dimensional tensor product space over the finite field with two elements, where \$A = e_1 \otimes e_2 \otimes ...