Articles of multilinear algebra

Associativity of Tensor Product

I have a doubt on the associativity of the tensor product. I know that the tensor product of vector spaces is an associative operation up to a linear isomorphism and I’m just trying to prove that. My idea is: Let $V_1, \dots ,V_p$ be vector spaces over the same field $\mathbb{K}$ and let $k, r […]

Correct definition of bilinear(multilinear) maps over noncommutative rings

I wonder about the following: Let $R$ be a noncommutative ring with $\alpha, \beta \in R$ such that $\alpha \beta \neq \beta \alpha$. Let $M,N,P$ be left-$R$-modules. Why does it make problems to define a map $B: M \times N \to P$ to be bilinear if $B(\lambda m,n) = \lambda B(m,n)$ and analogously for the […]

Why is the the $k$-th derivative a symmetric multilinear map?

I am having trouble understanding, why the $k$-th derivative of a map $F\colon\mathbb R^n \to\mathbb R^m$ is a symmetric multilinear map for each $x$ in $\mathbb R^n$. Can you please explain which vectors this map accepts as input where multilinearity comes from ? Also, why is symmetry mentioned ? Thank you readingframe

Prove some properties of determinant by the universal property of the tensor product of vector spaces

I am looking forward to an application of the universal property of the tensor product of vector spaces. We DEFINE the determinant of a $n\times n$ matrix $A=(a_{ij})$ over a field $F$ by that for any fixed $i=1, 2, …, n$, $$\det{A}=\sum_{j=1}^{n}(-1)^{i+j}a_{ij} |A_{ij}|,$$ where $|A_{ij}|$ is the minor of $A$ at position $(i, j)$. The […]

Existence of isomorphism between tensor products.

In multilinear algebra many maps are usually proven to exist rather than simply defined. For example, commutativity is one such example. In the book I’m studying the author says: let $V_1,\dots,V_k$ be a collection of vector spaces over $K$, then if $\sigma \in S_k$ there is a linear isomorphism $$f_\sigma : V_1\otimes\cdots\otimes V_k\to V_{\sigma(1)}\otimes\cdots\otimes V_{\sigma(k)}$$ […]

Inner Products on Exterior Powers

Let $H$ is a real, $n$-dimensional vector space. Define $\varphi \colon \operatorname{GL}(H) \rightarrow \operatorname{GL}(\wedge^{k}H)$ by $A \mapsto \wedge^{k}A$ and $\psi_{\langle \cdot, \cdot \rangle} \colon \operatorname{GL}(\wedge^{k}H) \rightarrow T$ given by $J \mapsto\langle J(\cdot),J(\cdot)⟩$. Here, $\langle \cdot, \cdot \rangle$ is some inner product on $H$, $$ T = \{g \colon \wedge^{k}H\times \wedge^{k}H \, | \, g \text{ […]

Do any non-combinatorial proofs of the elementary properties of wedge products exist?

The wege product, an operation defined between two alternating tensors, has a number of elementary properties such as associativity, distributivity, etc. There are many proofs of these properties e.g., see Analysis on Manifolds by Mukres or Topology, Geometry and Guage Fields by Naber. These proofs differ in the details but all ultimately rely on combinatorial […]

Proving Things About Rings Using Things About Vector Spaces

All rings below are assumed to be commutative and having an identity. $\newcommand{\bw}{\bigwedge}\newcommand{\R}{\mathbf R}\newcommand{\mc}{\mathcal}$ Consider the following problem: Problem 1. Let $M$ and $N$ be $n\times n$ matrices with entries from a field $F$ of characteristic $0$. Then $$\det(MN)=\det M\det N\tag{1}$$ The above can be neatly proved using exterior algebras (I have hidden the details […]

Tensors = matrices + covariance/contravariance?

I have read several topics on tensors but it is still not clear to me. Tensors are different from matrices because they contain additional information about how do they transform. To fully specify a $n\times n$ matrix, one need to specify its $n^2$ components. My question is: what do someone need to specify to fully […]

Rank $1$ bilinear form is a product of two linear functionals on a finite dimensional vector space.

Let $\mathbb{f}$ be a non-zero bilinear form on a finite dimensional vector sppace $V.$ Then have to show that $\mathbb{f}$ can be expressed as a product of two linear functionals i.e., $\mathbb{f}(\alpha, \beta)=L_1(\alpha)L_2(\beta)$ for $L_i \in V^*$ iff $\mathbb{f}$ has rank $1.$ I proved that if $\mathbb{f}$ is product of two linear functional then its […]