duality with respect to a non-degenerate bilinear form

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

=

{

v

1

,

…

,

v

n

}

be a basis of

V

and let

C

=

{

w

1

,

…

,

w

n

}

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

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

∈

V

and

w

∈

W

. By Theorem 1, we have

W

≅

V

∗

. Let

ℬ

=

{

v

1

,

…

,

v

n

}

be a basis for

V

and let

𝒞

=

{

w

1

,

…

,

w

n

}

be a basis for

W

which
corresponds to the dual basis of

V

∗

via the

isomorphism

W

≅

V

∗

. Then

B

⁢

(

v

i

,

w

j

)

=

1

for

i

=

j

and equal to

0

otherwise. Let

A

=

(

α

i

⁢

j

)

be the
matrix of

f

with respect to

ℬ

. Then

f

⁢

(

v

j

)

=

∑

i

=

1

n

α

i

⁢

j

⁢

v

i

.

Let

A

′

=

(

β

i

⁢

j

)

be the matrix of

g

with respect to

𝒞

so that

g

⁢

(

w

j

)

=

∑

i

β

i

⁢

j

⁢

w

i

. We will
show that

A

′

=

A

T

, the transpose of

A

. Indeed:

B

⁢

(

f

⁢

(

v

j

)

,

w

k

)

=

B

⁢

(

∑

i

α

i

⁢

j

⁢

v

i

,

w

k

)

=

α

k

⁢

j

and also

B

⁢

(

f

⁢

(

v

j

)

,

w

k

)

=

B

⁢

(

v

j

,

g

⁢

(

w

k

)

)

=

B

⁢

(

v

j

,

∑

i

β

i

⁢

k

⁢

w

i

)

=

β

j

⁢

k

.

Therefore

β

j

⁢

k

=

α

k

⁢

j

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

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