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:V×W→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∗ be the dual space of V, i.e. W is the vector space formed by all linear transformations V→F. Let B:V×V∗→F be defined by B(v,f)=f(v) for all v∈V and all f:V→F in V∗. Then B is a non-degenerate bilinear form and V and V∗ are dual with respect to B.
Definition 2.
Let f:V→V and g:W→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∈V and w∈W.
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 w∈W one can define a linear form V→F 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×W→F. Then there exist canonical isomorphisms V≅W∗ and W≅V∗ given by
W→V∗,w↦(v↦B(v,w));V→W∗,v↦(w↦B(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×W→F. Moreover, suppose f:V→V and g:W→W 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 W≅V∗ 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 v∈V and w∈W. By Theorem 1, we have W≅V∗. Let
ℬ={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 isomorphism W≅V∗. Then B(vi,wj)=1
for i=j and equal to 0 otherwise. Let A=(αij) be the
matrix of f with respect to ℬ. Then
f(vj)=n∑i=1αijvi. |
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 |
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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 |