# duality with respect to a non-degenerate bilinear form

###### Definition 1.

Let $V$ and $W$ be finite dimensional vector spaces^{} over a field $F$ and let $B\mathrm{:}V\mathrm{\times}W\mathrm{\to}F$ be a non-degenerate bilinear form. Then we say that $V$ and $W$ are dual with respect to $B$.

###### Example 1.

Let $V$ be a finite dimensional vector space and let $W={V}^{\ast}$ be the dual space of $V$, i.e. $W$ is the vector space formed by all linear transformations $V\to F$. Let $B:V\times {V}^{\ast}\to F$ be defined by $B(v,f)=f(v)$ for all $v\in V$ and all $f:V\to F$ in ${V}^{\ast}$. Then $B$ is a non-degenerate bilinear form and $V$ and ${V}^{\ast}$ are dual with respect to $B$.

###### Definition 2.

Let $f\mathrm{:}V\mathrm{\to}V$ and $g\mathrm{:}W\mathrm{\to}W$ be linear transformations. We say that $f$ and $g$ are transposes^{} of each other with respect to $B$ if

$$B(f(v),w)=B(v,g(w))$$ |

for all $v\mathrm{\in}V$ and $w\mathrm{\in}W$.

The reasons why the terms “dual” and “transpose” are used are explained in the following theorems (here ${V}^{\ast}$ denotes the dual vector space of $V$). Notice that for a fixed element $w\in W$ one can define a linear form^{} $V\to F$ which sends $v$ to $B(v,w)$.

###### Theorem 1.

Let $V\mathrm{,}W$ be finite dimensional vector spaces over $F$ which are dual with respect to a non-degenerate bilinear form $B\mathrm{:}V\mathrm{\times}W\mathrm{\to}F$. Then there exist canonical isomorphisms $V\mathrm{\cong}{W}^{\mathrm{\ast}}$ and $W\mathrm{\cong}{V}^{\mathrm{\ast}}$ given by

$$W\to {V}^{\ast},w\mapsto (v\mapsto B(v,w));V\to {W}^{\ast},v\mapsto (w\mapsto B(v,w)).$$ |

###### Theorem 2.

Let $V\mathrm{,}W$ be finite dimensional vector spaces over $F$ which are dual with respect to a non-degenerate bilinear form $B\mathrm{:}V\mathrm{\times}W\mathrm{\to}F$. Moreover, suppose $f\mathrm{:}V\mathrm{\to}V$ and $g\mathrm{:}W\mathrm{\to}W$ are transposes of each other with respect to $B$. Let $\mathrm{B}\mathrm{=}\mathrm{\{}{v}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{v}_{n}\mathrm{\}}$ be a basis of $V$ and let $\mathrm{C}\mathrm{=}\mathrm{\{}{w}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{w}_{n}\mathrm{\}}$ be the basis of $W$ which maps to the dual basis of $\mathrm{B}$ via the isomporphism $W\mathrm{\cong}{V}^{\mathrm{\ast}}$ defined in the previous theorem. If $A$ is the matrix of $f$ in the basis $\mathrm{B}$ then the matrix of $g$ in the basis $\mathrm{C}$ is ${A}^{T}$, the transpose matrix of $A$.

###### Proof of Theorem 2..

Let $V$ and $W$ be dual with respect to a non-degenerate bilinear form $B$ and let $f$ and $g$ be transposes of each other, also with respect to $B$ so that:

$$B(f(v),w)=B(v,g(w))$$ |

for all $v\in V$ and $w\in W$. By Theorem 1, we have $W\cong {V}^{\ast}$. Let
$\mathcal{B}=\{{v}_{1},\mathrm{\dots},{v}_{n}\}$ be a basis for $V$ and let
$\mathcal{C}=\{{w}_{1},\mathrm{\dots},{w}_{n}\}$ be a basis for $W$ which
corresponds to the dual basis of ${V}^{\ast}$ via the isomorphism^{} $W\cong {V}^{\ast}$. Then $B({v}_{i},{w}_{j})=1$
for $i=j$ and equal to $0$ otherwise. Let $A=({\alpha}_{ij})$ be the
matrix of $f$ with respect to $\mathcal{B}$. Then

$$f({v}_{j})=\sum _{i=1}^{n}{\alpha}_{ij}{v}_{i}.$$ |

Let ${A}^{\prime}=({\beta}_{ij})$ be the matrix of $g$ with respect to $\mathcal{C}$ so that $g({w}_{j})={\sum}_{i}{\beta}_{ij}{w}_{i}$. We will show that ${A}^{\prime}={A}^{T}$, the transpose of $A$. Indeed:

$$B(f({v}_{j}),{w}_{k})=B(\sum _{i}{\alpha}_{ij}{v}_{i},{w}_{k})={\alpha}_{kj}$$ |

and also

$$B(f({v}_{j}),{w}_{k})=B({v}_{j},g({w}_{k}))=B({v}_{j},\sum _{i}{\beta}_{ik}{w}_{i})={\beta}_{jk}.$$ |

Therefore ${\beta}_{jk}={\alpha}_{kj}$ for all $k$ and $j$, as desired. ∎

Title | duality with respect to a non-degenerate bilinear form |
---|---|

Canonical name | DualityWithRespectToANondegenerateBilinearForm |

Date of creation | 2013-03-22 16:23:02 |

Last modified on | 2013-03-22 16:23:02 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 8 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 15A99 |

Related topic | BilinearForm |

Related topic | PolaritiesAndForms |