# Questions tagged [multilinear-algebra]

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

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### Grounding the concept of a Free Vector space of the cartesian product of two vector spaces

$\def\tv{\tilde{v}}$ $\def\tw{\tilde{w}}$ $\def\F{\mathbb{F}}$ In constructing the Tensor Product of two, finite-dimensional, vector spaces $V,W$ over a field $\F$ it is common to start from the Free ...
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### Werner Greub's formulation of the Universal Property of the Tensor Product

$\def\id{\operatorname{id}}$ $\def\Im{\operatorname{Im}}$In Section 1.4 of Multilinear Algebra Werner Greub starts with a bilinear map $\otimes: E \times F \rightarrow T$ where $E,F,T$ are vector ...
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### Inversion of a matrix equation

Is there a general way to invert (solve for $u$) this? $$\sum_{ij}R_{ijk}a_iu_j = -x_k$$ With $a,u,x \in \mathbb{R}^N$. $R_{ijk}$ is symmetric in the last two indices. So really I'm trying to invert ...
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### Comultiplication on the tensor algebra

Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal ...
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### Induced change of basis on a (p,q) tensor

I'm struggling to simplify the last step of a $(p,q)$ tensor and how its components change with a linear change of basis on the associated vector space. So far I have: Given a vector space $V$ over ...
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### Tensor confusion: scalar $\otimes$ vector = OK?...

I am trying to understand tensors and one particular question have caused me a great deal of confusion. The particular example with the metric tensor below is an attempt to highlight where my ...
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### On the tensor product

Let us start with $V,W$ to $R$-modules. In order to define their tensor product we first introduce something called the free R-module $Free(V \times W)$ whose basis is given by the set of all ordered ...
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### What is the different meaning of $k$-tensor and tensor of type $(p,q)$?

Let $V$ be a $n$ dimensional vector space over $\mathbb R$ and $k,p,q\in\mathbb N$. A $k$-tensor is a $k$-multilinear functional on $V$, that is a map \begin{align} f:&& V^k&\...
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### About $\operatorname{Alt}(\varphi_{i_1}\otimes\cdots\otimes\varphi_{i_k})$ ("Calculus on Manifolds" by Michael Spivak)
I am reading "Calculus on Manifolds" by Michael Spivak. The author wrote as follows: Since each $\operatorname{Alt}(\varphi_{i_1}\otimes\cdots\otimes\varphi_{i_k})$ is a constant times one ...