duality with respect to a non-degenerate bilinear form


Definition 1.

Let V and W be finite dimensional vector spacesMathworldPlanetmath over a field F and let B:V×WF 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 be the dual space of V, i.e. W is the vector space formed by all linear transformations VF. Let B:V×VF be defined by B(v,f)=f(v) for all vV and all f:VF in V. Then B is a non-degenerate bilinear form and V and V are dual with respect to B.

Definition 2.

Let f:VV and g:WW be linear transformations. We say that f and g are transposesMathworldPlanetmath of each other with respect to B if

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

for all vV and wW.

The reasons why the terms “dual” and “transpose” are used are explained in the following theorems (here V denotes the dual vector space of V). Notice that for a fixed element wW one can define a linear formPlanetmathPlanetmath VF which sends v to B(v,w).

Theorem 1.

Let V,W be finite dimensional vector spaces over F which are dual with respect to a non-degenerate bilinear form B:V×WF. Then there exist canonical isomorphisms VW and WV given by

WV,w(vB(v,w));VW,v(wB(v,w)).
Theorem 2.

Let V,W be finite dimensional vector spaces over F which are dual with respect to a non-degenerate bilinear form B:V×WF. Moreover, suppose f:VV and g:WW are transposes of each other with respect to B. Let B={v1,,vn} be a basis of V and let C={w1,,wn} be the basis of W which maps to the dual basis of B via the isomporphism WV defined in the previous theorem. If A is the matrix of f in the basis B then the matrix of g in the basis C is AT, 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 vV and wW. By Theorem 1, we have WV. Let ={v1,,vn} be a basis for V and let 𝒞={w1,,wn} be a basis for W which corresponds to the dual basis of V via the isomorphismPlanetmathPlanetmath WV. Then B(vi,wj)=1 for i=j and equal to 0 otherwise. Let A=(αij) be the matrix of f with respect to . Then

f(vj)=i=1nαijvi.

Let A=(βij) be the matrix of g with respect to 𝒞 so that g(wj)=iβijwi. We will show that A=AT, the transpose of A. Indeed:

B(f(vj),wk)=B(iαijvi,wk)=αkj

and also

B(f(vj),wk)=B(vj,g(wk))=B(vj,iβikwi)=βjk.

Therefore βjk=αkj 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