Multilinear mapping/Alternating/Introduction/Section

Definition

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ and ${}W$ be vector spaces over ${}K$. A mapping

$\Phi \colon V_{1}\times \cdots \times V_{n}\longrightarrow W$

is called multilinear if, for every ${}i\in \{1,\ldots ,n\}$ and every ${}(n-1)$-tuple ${}(v_{1},\ldots ,v_{i-1},v_{i+1},\ldots ,v_{n})$ with ${}v_{j}\in V_{j}$, the induced mapping

$V_{i}\longrightarrow W,v_{i}\longmapsto \Phi (v_{1},\ldots ,v_{i-1},v_{i},v_{i+1},\ldots ,v_{n}),$

is

${}K$-linear.

For ${}n=2$, this property is called bilinear. For example, the multiplikation in a field ${}K$, that is, the mapping

$K\times K\longrightarrow K,(x,y)\longmapsto xy,$

is bilinear. Also, for a ${}K$-vector space ${}V$ and its dual space ${}{V}^{*}$, the evaluation mapping

$V\times {V}^{*}\longrightarrow K,(v,f)\longmapsto f(v),$

is bilinear.

Lemma

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ and ${}W$ be vector spaces over ${}K$. Let

$\Phi \colon V_{1}\times \cdots \times V_{n}\longrightarrow W$

be a multilinear mapping, and let ${}v_{1j},\ldots ,v_{m_{j}j}\in V_{j}$ and ${}a_{ij}\in K$. Then

$\Phi {\left(\sum _{i=1}^{m_{1}}a_{i1}v_{i1},\ldots ,\sum _{i=1}^{m_{n}}a_{in}v_{in}\right)}=\sum _{(i_{1},\ldots ,i_{n})\in \{1,\ldots ,m_{1}\}\times \cdots \times \{1,\ldots ,m_{n}\}}a_{i_{1}1}\cdots a_{i_{n}n}\Phi {\left(v_{i_{1}1},\ldots ,v_{i_{n}n}\right)}\,.$

Proof

See exercise.

$\Box$

Definition

Let ${}K$ be a field, let ${}V$ and ${}W$ denote ${}K$-vector spaces, and let ${}n\in \mathbb {N}$. A multilinear mapping

$\Phi \colon V^{n}=\underbrace {V\times \cdots \times V} _{n{\text{-mal}}}\longrightarrow W$

is called alternating if the following holds: Whenever in ${}v=(v_{1},\ldots ,v_{n})$, two entries are identical, that is ${}v_{i}=v_{j}$ for a pair ${}i\neq j$, then

${}\Phi (v)=0\,.$

For an alternating mapping, there is only one vector space occurring several times in the product on the left.

Lemma

Let ${}K$ be a field, let ${}V$ and ${}W$ denote ${}K$-vector spaces, and let ${}n\in \mathbb {N}$. Suppose that

$\Phi \colon V^{n}=\underbrace {V\times \cdots \times V} _{n{\text{-mal}}}\longrightarrow W$

is an alternating mapping. Then

$\Phi (v_{1},\ldots ,v_{r-1},v_{r},v_{r+1},\ldots ,v_{s-1},v_{s},v_{s+1},\ldots ,v_{n})=-\Phi (v_{1},\ldots ,v_{r-1},v_{s},v_{r+1},\ldots ,v_{s-1},v_{r},v_{s+1},\ldots ,v_{n})\,.$

This means that if we swap two vectors, them the sign is changing.

Proof

Due to the definition of alternating and fact, we have

${}{\begin{aligned}0&=\Phi (v_{1},\ldots ,v_{r-1},v_{r}+v_{s},v_{r+1},\ldots ,v_{s-1},v_{r}+v_{s},v_{s+1},\ldots ,v_{n})\\&=\Phi (v_{1},\ldots ,v_{r-1},v_{r},v_{r+1},\ldots ,v_{s-1},v_{s},v_{s+1},\ldots ,v_{n})+\Phi (v_{1},\ldots ,v_{r-1},v_{s},v_{r+1},\ldots ,v_{s-1},v_{r},v_{s+1},\ldots ,v_{n}).\end{aligned}}$

$\Box$