# Bilinear form/Introduction/Section

Real inner products are positive definite symmetric bilinear forms. In the following, we discuss bilinear forms in general. Beside inner products, there are the Hesse-forms, which are important in higher dimensional analysis in order to determine extrema, and the Minkowski-forms, which are used to describe special relativity theory.

## Definition

Let $K$ be a field, and let $V$ denote a $K$-vector space. A mapping
    $V\times V\longrightarrow K,(v,w)\longmapsto \left\langle v,w\right\rangle ,$
is called a bilinear form, if, for all $v\in V$, the induced mappings
    $V\longrightarrow K,w\longmapsto \left\langle v,w\right\rangle ,$
and for all $w\in V$, the induced mappings
    $V\longrightarrow K,v\longmapsto \left\langle v,w\right\rangle ,$
are
$K$
Bilinear simply means being multilinear in two components. An extreme example is the zero form, which assigns to every pair the value $0$. It is easy to describe many different bilinear forms on $K^{n}$.

## Example

Let $V=K^{n}$, and let $a_{ij}\in K$ denote fixed numbers, for $1\leq i,j\leq n$. Then, the assignment
    $({\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}},{\begin{pmatrix}y_{1}\\\vdots \\y_{n}\end{pmatrix}})\longmapsto \Psi (x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})=\sum _{i,j}a_{ij}x_{i}y_{j}$
is a bilinear form. In case
    $a_{ij}=0\,$
for all $i,j$, this is the zero form; in case
    $a_{ij}=\delta _{ij}={\begin{cases}1,{\text{ if }}i=j\,,\\0{\text{ else}}\,,\end{cases}}\,$
we have the standard inner product (where the expression makes sense for every field; the property of being positive definite does only make sense for an ordered field). In case $n=4$ and
    $\Psi (x_{1},\ldots ,x_{4},y_{1},\ldots ,y_{4})=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}\,,$
we talk about a Minkowski-form. For $n=2$ and
    $\Psi (x_{1},x_{2},y_{1},y_{2})=x_{1}y_{2}-x_{2}y_{1}\,,$
this is the determinant in the two-dimensional case.
An important property of a bilinear form (which inner products fulfill) is formulated in the next definition.

## Definition

Let $K$ be a field, and let $V$ denote a $K$-vector space. A bilinear form
    $V\times V\longrightarrow K,(v,w)\longmapsto \left\langle v,w\right\rangle ,$
is called nondegenerate, if for every $v\in V$, $v\neq 0$, the induced mapping
    $V\longrightarrow K,w\longmapsto \left\langle v,w\right\rangle ,$
and, for every $w\in V$, $w\neq 0$, the induced mapping
    $V\longrightarrow K,v\longmapsto \left\langle v,w\right\rangle ,$
is not the
We will prove in fact, for a vector space endowed with a nondegenerate bilinear form, the existence of a natural bijective relation between vectors and linear forms. This holds in particular for inner products. In general, there is a strong relation between bilinear forms and linear mappings to the dual space.

## Lemma

Let $K$ be a field, let $V$ denote a $K$-vector space with its dual space ${V}^{*}$. Let
    $\Theta \colon V\longrightarrow {V}^{*}$
be a linear mapping. Then
    $\Psi (u,v)=(\Theta (u))(v)\,$
defines a bilinear form
$V$

### Proof

Since $\Theta (u)\in {V}^{*}$, the evaluation at a vector $v\in V$ yields an element of the base field. The linearity in the second component rests on the fact that $\Theta (u)$ belongs to the dual space. The linearity in the first component rest on the linearity of $\Theta$.
$\Box$