aarjaneiro/planetmath huggingface dataset:
alternating form

alternating form

A

bilinear form

A

on a

vector space

V

(over a field

k

) is called an

alternating form

if for all

v

∈

V

,

A

⁢

(

v

,

v

)

=

0

.

Since for any

u

,

v

∈

V

,

0

=

A

⁢

(

u

+

v

,

u

+

v

)

=

A

⁢

(

u

,

u

)

+

A

⁢

(

u

,

v

)

+

A

⁢

(

v

,

u

)

+

A

⁢

(

v

,

v

)

=

A

⁢

(

u

,

v

)

+

A

⁢

(

v

,

u

)

,

we see that

A

⁢

(

u

,

v

)

=

-

A

⁢

(

v

,

u

)

. So an alternating form is automatically a

anti-symmetric

, or

skew symmetric

form. The converse is true if the

characteristic

of

k

is not

2

.

Let

V

be a two dimensional vector space over

k

with an alternating form

A

. Let

{

e

1

,

e

2

}

be a basis for

V

. The matrix associated with

A

looks like

(

A

⁢

(

e

1

,

e

1

)

A

⁢

(

e

1

,

e

2

)

A

⁢

(

e

2

,

e

1

)

A

⁢

(

e

2

,

e

2

)

)

=

r

⁢

(

0

1

-

1

0

)

=

r

⁢

S

,

where

r

=

A

⁢

(

e

1

,

e

2

)

. The

skew symmetric matrix

S

has the property that its

diagonal

entries are all

0

.

S

is called the

2

×

2

alternating

or

symplectic matrix

.

A

is called

non-singular

or

non-degenerate

if there exist a vectors

u

,

v

∈

V

such that

A

⁢

(

u

,

v

)

≠

0

.

u

,

v

are necessarily non-zero. Note that the associated matrix

r

⁢

S

is non-singular iff

r

≠

0

iff

A

is non-singular.

In the two dimensional vector space case above, if

A

is non-singular, we can re-scale the basis elements so that

r

=

1

. This means that the matrix associated with

A

is the alternating matrix. A two-dimensional vector space which carries a non-singular alternating form is sometimes called an

alternating

or

symplectic hyperbolic plane

. Some authors also call it simply a

hyperbolic plane

. But here on PlanetMath, we will reserve the shorter name for its cousin in the category of

quadratic spaces

. Let’s denote an alternating hyperbolic plane by

𝒜

.

Remark.

In general, it can be shown that if

V

is an

n

-dimensional vector space equipped with a non-singular alternating form

A

, then

V

can be written as an

orthogonal direct sum

of the alternating hyperbolic planes

𝒜

. In other words, the associated matrix for

A

has the block form

(

S

𝟎

⋯

𝟎

𝟎

S

⋯

𝟎

⋮

⋮

⋱

⋮

𝟎

𝟎

⋯

S

)

,

where

⁢

𝟎

=

(

0

0

0

0

)

.

Furthermore,

n

is even.

V

is called a

symplectic vector space

.

Title

alternating form

Canonical name

AlternatingForm

Date of creation

2013-03-22 15:42:17

Last modified on

2013-03-22 15:42:17

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A63

Synonym

alternate form

Synonym

alternating

Synonym

symplectic hyperbolic plane

Related topic

SymplecticVectorSpace

Related topic

EverySymplecticManifoldHasEvenDimension

Defines

alternating hyperbolic plane

Generated on Fri Feb 9 18:29:39 2018 by

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area of polygon

area of polygon

Let the vertices of a ()

polygon

be

(

x

1

,

y

1

)

,

(

x

2

,

y

2

)

,

…

,

(

x

n

,

y

n

)

,  enumerated in when gone round the polygon anticlockwise. The area of the polygon is equal to

1

2

⁢

(

|

x

1

x

2

y

1

y

2

|

+

|

x

2

x

3

y

2

y

3

|

+

…

+

|

x

n

-

1

x

n

y

n

-

1

y

n

|

+

|

x

n

x

1

y

n

y

1

|

)

.

References

1

E. Lindelöf:

Johdatus korkeampaan analyysiin

. Neljäs painos.  Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1956).

Title

area of polygon

Canonical name

AreaOfPolygon

Date of creation

2013-03-22 17:32:57

Last modified on

2013-03-22 17:32:57

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Result

Classification

msc 15A63

Classification

msc 51M25

Related topic

Area

Related topic

Determinant2

Related topic

CentreOfMassOfPolygon

Generated on Fri Feb 9 18:29:41 2018 by

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Cauchy-Schwarz inequality

Cauchy-Schwarz inequality

Let

V

be a

vector space

where an

inner product

⟨

,

⟩

has been defined.
Such spaces can be given also a norm by defining

∥

x

∥

=

⟨

x

,

x

⟩

.

Then in such a space the

Cauchy-Schwarz inequality

holds:

|

⟨

v

,

w

⟩

|

≤

∥

v

∥

⁢

∥

w

∥

for any

v

,

w

∈

V

. That is, the

modulus

(since it might as well be a

complex number

) of the inner product for two given vectors is less or equal than the product of their norms. Equality happens if and only if the two vectors are

linearly dependent

.

A very special case is when

V

=

ℝ

n

and the inner product is the

dot product

defined as

⟨

v

,

w

⟩

=

v

t

⁢

w

and usually denoted as

v

⋅

w

and the resulting norm is the

Euclidean norm

.

If

𝐱

=

(

x

1

,

x

2

,

…

,

x

n

)

and

𝐲

=

(

y

1

,

y

2

,

…

,

y

n

)

the Cauchy-Schwarz inequality becomes

|

𝐱

⋅

𝐲

|

=

|

x

1

⁢

y

1

+

x

2

⁢

y

2

+

⋯

+

x

n

⁢

y

n

|

≤

x

1

2

+

x

2

2

+

⋯

+

x

n

2

⁢

y

1

2

+

y

2

2

+

⋯

+

y

n

2

=

∥

𝐱

∥

⁢

∥

𝐲

∥

,

which implies

(

x

1

⁢

y

1

+

x

2

⁢

y

2

+

⋯

+

x

n

⁢

y

n

)

2

≤

(

x

1

2

+

x

2

2

+

⋯

+

x

n

2

)

⁢

(

y

1

2

+

y

2

2

+

⋯

+

y

n

2

)

Notice that in this case

inequality

holds even if the modulus on the middle term (which is a

real number

) is not used.

Cauchy-Schwarz inequality is also a special case of Hölder inequality.
The inequality arises in lot of fields, so it is known under several other names as
Bunyakovsky inequality or

Kantorovich inequality

. Another form that arises often is Cauchy-Schwartz inequality but this is a misspelling since the inequality is named after

http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Schwarz.html

Hermann Amandus Schwarz (1843–1921).

This inequality is

similar

to the

triangle inequality

, talking about products instead of sums:

∥

𝐱

+

𝐲

∥

≤

∥

𝐱

∥

+

∥

𝐲

∥

⁢

triangle inequality

|

𝐱

⋅

𝐲

|

≤

∥

𝐱

∥

⋅

∥

𝐲

∥

⁢

CS inequality

Title

Cauchy-Schwarz inequality

Canonical name

CauchySchwarzInequality

Date of creation

2013-03-22 12:14:46

Last modified on

2013-03-22 12:14:46

Owner

drini (3)

Last modified by

drini (3)

Numerical id

20

Author

drini (3)

Entry type

Theorem

Classification

msc 15A63

Synonym

Kantorovich inequality

Synonym

Bunyakovsky inequality

Synonym

Schwarz inequality

Synonym

Cauchy inequality

Synonym

CBS inequality

Related topic

InnerProduct

Related topic

InnerProductSpace

Related topic

DotProduct

Related topic

NormedVectorSpace

Related topic

VectorNorm

Related topic

CauchySchwartzInequality

Related topic

HolderInequality

Related topic

CauchySchwarzInequality

Related topic

VectorPnorm

Defines

Cauchy-Schwartz inequality

Generated on Fri Feb 9 18:29:43 2018 by

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Euclidean vector space

Euclidean vector space

1

Definition

The term

Euclidean vector space

is synonymous with

finite-dimensional

, real,

positive definite

,

inner product space

. The canonical example is

ℝ

n

, equipped with the usual

dot product

. Indeed, every Euclidean vector space

V

is

isomorphic

to

ℝ

n

, up to a choice of

orthonormal basis

of

V

. As well, every Euclidean vector space

V

carries a natural

metric space

structure given by

d

⁢

(

u

,

v

)

=

⟨

u

-

v

,

u

-

v

⟩

,

u

,

v

∈

V

.

2

Remarks.

•

An analogous object with

complex numbers

as the

base field

is called a

unitary space

.

•

Dropping the assumption of finite-dimensionality we arrive at the class of real

pre-Hilbert spaces

.

•

If we drop the

inner product

and the

vector space

structure, but retain the metric space structure, we arrive at the notion of a

Euclidean space

.

Title

Euclidean vector space

Canonical name

EuclideanVectorSpace

Date of creation

2013-03-22 15:38:24

Last modified on

2013-03-22 15:38:24

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A63

Related topic

InnerProductSpace

Related topic

UnitarySpace

Related topic

PositiveDefinite

Related topic

EuclideanDistance

Related topic

Vector

Related topic

EuclideanVectorSpace

Generated on Fri Feb 9 18:29:45 2018 by

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every orthonormal set is linearly independent

every orthonormal set is linearly independent

Theorem

: An

orthonormal set

of vectors in an

inner product space

is

linearly independent

.

Proof.

We denote by

⟨

⋅

,

⋅

⟩

the

inner product

of

L

. Let

S

be an orthonormal set of vectors.
Let us first consider the case when

S

is finite, i.e.,

S

=

{

e

1

,

…

,

e

n

}

for some

n

.
Suppose

λ

1

⁢

e

1

+

⋯

+

λ

n

⁢

e

n

=

0

for some scalars

λ

i

(belonging to the field on the
underlying

vector space

of

L

). For a fixed

k

in

1

,

…

,

n

,
we then have

0

=

⟨

e

k

,

0

⟩

=

⟨

e

k

,

λ

1

⁢

e

1

+

⋯

+

λ

n

⁢

e

n

⟩

=

λ

1

⁢

⟨

e

k

,

e

1

⟩

+

⋯

+

λ

n

⁢

⟨

e

k

,

e

n

⟩

=

λ

k

,

so

λ

k

=

0

, and

S

is linearly independent.
Next, suppose

S

is

infinite

(

countable

or

uncountable

). To prove
that

S

is linearly independent, we need to show that
all finite subsets of

S

are linearly independent. Since any
subset of an orthonormal set is also orthonormal, the infinite case
follows from the finite case.

□

Title

every orthonormal set is linearly independent

Canonical name

EveryOrthonormalSetIsLinearlyIndependent

Date of creation

2013-03-22 13:33:48

Last modified on

2013-03-22 13:33:48

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 15A63

Generated on Fri Feb 9 18:29:47 2018 by

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Hasse-Minkowski theorem

Hasse-Minkowski theorem

The

Hasse-Minkowski theorem

is a classical example of the

Hasse principle

.

Let

F

be a

global field

, i.e. a

number field

or a

rational function field

over a

finite field

of

characteristic

not

2

,

X

a finite dimensional

vector space

over

F

and

ϕ

a

regular quadratic form

over

X

.

A regular quadratic form

ϕ

over

X

is a

quadratic form

such that for every

x

≠

0

in

X

there is a

y

in

X

with

b

⁢

(

x

,

y

)

≠

0

. Here

b

⁢

(

x

,

y

)

=

1

2

⁢

(

q

⁢

(

x

+

y

)

-

q

⁢

(

x

)

-

q

⁢

(

y

)

)

is the associated

bilinear form

.

To every completion

F

v

of

F

with respect to a nontrivial

valuation

v

we assign the vector space

X

v

:=

F

v

⊗

F

X

and the

induced

quadratic form

ϕ

v

on

X

v

.

A quadratic form

ϕ

over

X

is an

isotropic quadratic form

if there is a nonzero vector

x

∈

X

with

ϕ

⁢

(

x

)

=

0

.

The Hasse-Minkowski theorem can now be stated as:

Theorem 1

A regular quadratic form

ϕ

over a global field

F

is isotropic if and only if every completion

ϕ

v

is isotropic, where

v

runs through the nontrivial valuations of

F

.

The case of

ℚ

was first proved by Minkowski. It can be proved using the

Hilbert symbol

and Dirichlet’s theorem on primes in arithmetic progressions.

The general case was proved by Hasse. It can be proved using two local-global principles of

class field theory

, namely the

Hasse norm theorem

:
For a cyclic

field extension

E

/

F

of global fields an element

a

∈

F

is a norm of

E

/

F

and only if it is a norm of

E

v

/

F

v

for every valuation

v

of

E

.

and the

Global square theorem

:
An element

a

of a global field

F

is a square if and only if it is a square in every

F

v

.

Title

Hasse-Minkowski theorem

Canonical name

HasseMinkowskiTheorem

Date of creation

2013-03-22 15:19:47

Last modified on

2013-03-22 15:19:47

Owner

SirJective (9710)

Last modified by

SirJective (9710)

Numerical id

8

Author

SirJective (9710)

Entry type

Theorem

Classification

msc 15A63

Classification

msc 14G05

Related topic

HassePrinciple

Related topic

QuadraticForm

Defines

regular quadratic form

Generated on Fri Feb 9 18:29:55 2018 by

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Hermitian form over a division ring

Hermitian form over a division ring

Let

D

be a division ring admitting an

involution

(

http://planetmath.org/Involution2

)

*

. Let

V

be a

vector space

over

D

. A

Hermitian form

over

D

is a function from

V

×

V

to

D

, denoted by

(

⋅

,

⋅

)

with the following

properties

, for any

v

,

w

∈

V

and

d

∈

D

:

1.

(

⋅

,

⋅

)

is

additive

in each of its arguments,

2.

(

d

⁢

u

,

v

)

=

d

⁢

(

u

,

v

)

,

3.

(

u

,

d

⁢

v

)

=

(

u

,

v

)

⁢

d

*

,

4.

(

u

,

v

)

=

(

v

,

u

)

*

.

Note that if the Hermitian form

(

⋅

,

⋅

)

is non-trivial and if

*

is the identity on

D

, then

D

is a field and

(

⋅

,

⋅

)

is just a

symmetric bilinear form

.

If we replace the last condition by

(

u

,

v

)

=

-

(

v

,

u

)

*

, then

(

⋅

,

⋅

)

over

D

is called a

skew Hermitian form

.

Remark.

Every skew Hermitian form over a division ring induces a Hermitian form and vice versa.

Title

Hermitian form over a division ring

Canonical

name

HermitianFormOverADivisionRing

Date of creation

2013-03-22 15:41:04

Last modified on

2013-03-22 15:41:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A63

Defines

Hermitian form

Defines

skew Hermitian form

Generated on Fri Feb 9 18:29:58 2018 by

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isotropic quadratic space

isotropic quadratic space

A vector

v

(an element of

V

) in a

quadratic space

(

V

,

Q

)

is

isotropic

if

1.

v

≠

0

and

2.

Q

⁢

(

v

)

=

0

.

Otherwise, it is called

anisotropic

. A quadratic space

(

V

,

Q

)

is isotropic if it contains an

isotropic vector

. Otherwise, it is anisotropic. A quadratic space

(

V

,

Q

)

is

totally isotropic

if every one of its non-zero vector is isotropic, or that

Q

⁢

(

V

)

=

0

.

Similarly, an

isotropic quadratic form

is one which has a non-trivial kernel, or that there exists a vector

v

such that

Q

⁢

(

v

)

=

0

. The definitions for that of an

anisotropic quadratic form

and that of a

totally isotropic quadratic form

should now be clear from the above discussion (anisotropic:

ker

⁡

(

Q

)

=

0

; totally isotropic:

ker

⁡

(

Q

)

=

V

).

Examples.

•

Consider the

quadratic form

Q

⁢

(

x

,

y

)

=

x

2

+

y

2

in the

vector space

ℝ

2

over the reals. It is clearly anisotropic since there are no

real numbers

a

,

b

not both

0

, such that

a

2

+

b

2

=

0

.

•

However, the same form is isotropic in

ℂ

2

over

ℂ

, since

1

2

+

i

2

=

0

; the

complex numbers

are

algebraically closed

.

•

Again, using the same form

x

2

+

y

2

, but in

ℝ

3

over the reals , we see that it is isotropic since the

z

term is missing, so that

Q

⁢

(

0

,

0

,

1

)

=

0

2

+

0

2

=

0

.

•

If we restrict

Q

to the

subspace

consisting of the

z

-axis (

x

=

y

=

0

) and call it

Q

z

, then

Q

z

is totally isotropic, and the

z

-axis is a totally isotropic subspace.

•

The quadratic form

Q

⁢

(

x

,

y

)

=

x

2

-

y

2

is clearly isotropic in any vector space over any field. In general, this is true if the

coefficients

of a

diagonal quadratic form

Q

consist of

1

,

-

1

,

0

(

0

is optional) and nothing else.

Title

isotropic quadratic space

Canonical name

IsotropicQuadraticSpace

Date of creation

2013-03-22 15:41:57

Last modified on

2013-03-22 15:41:57

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A63

Classification

msc 11E81

Related topic

QuadraticMap2

Related topic

QuadraticForm

Defines

isotropic vector

Defines

isotropic quadratic form

Defines

anisotropic vector

Defines

anisotropic quadratic form

Defines

anisotropic quadratic space

Defines

totally isotropic quadratic space

Defines

totally isotropic quadratic form

Generated on Fri Feb 9 18:30:00 2018 by

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left / right perpendicular

left / right perpendicular

Given a

sesquilinear form

b

:

V

×

V

→

k

over the field

k

, if

v

,

u

∈

V

such that

b

⁢

(

v

,

u

)

=

0

then we say

v

is

right perpendicular

to

u

and denote it

v

⊥

u

. Likewise

u

is

left perpendicular

to

v

and can be denoted by

u

⊤

v

.

By

definition

v

⟂

u

if and only if

u

⊤

v

. However,

v

⟂

u

need not imply

u

⟂

v

.

For example, let

V

=

k

⊕

k

and

b

⁢

(

(

v

1

,

v

2

)

,

(

u

1

,

u

2

)

)

=

v

1

⁢

u

2

. Then

b

⁢

(

(

0

,

1

)

,

(

1

,

0

)

)

=

0

so

(

0

,

1

)

⊥

(

1

,

0

)

, or equivalently,

(

1

,

0

)

⊤

(

0

,

1

)

.
However

b

⁢

(

(

1

,

0

)

,

(

0

,

1

)

)

=

1

≠

0

so

(

1

,

0

)

is not right perpendicular to

(

0

,

1

)

and

(

0

,

1

)

is not left perpendicular to

(

1

,

0

)

.

Title

left / right perpendicular

Canonical

name

LeftRightPerpendicular

Date of creation

2013-03-22 16:13:05

Last modified on

2013-03-22 16:13:05

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

7

Author

Algeboy (12884)

Entry type

Derivation

Classification

msc 15A63

Defines

left perpendicular

Defines

right perpendicular

Generated on Fri Feb 9 18:30:02 2018 by

LaTeXML
matrix representation of a bilinear form

matrix representation of a bilinear form

Given a

bilinear form

,

B

:

U

×

V

→

K

, we show how we can represent it with a matrix, with respect to a particular pair of bases for

U

and

V

Suppose

U

and

V

are

finite-dimensional

and we have chosen bases,

ℬ

1

=

{

e

1

,

…

}

and

ℬ

2

=

{

f

1

,

…

}

. Now we define the matrix

C

with entries

C

i

⁢

j

=

B

⁢

(

e

i

,

f

j

)

. This will be the matrix associated to

B

with respect to this basis as follows; If we write

x

,

y

∈

V

as

column vectors

in terms of the chosen bases, then check

B

⁢

(

x

,

y

)

=

x

T

⁢

C

⁢

y

. Further if we choose the corresponding dual bases for

U

∗

and

V

∗

then

C

and

C

T

are the corresponding matrices for

B

R

and

B

L

, respectively (in the sense of

linear maps

). Thus we see that a

symmetric bilinear form

is represented by a

symmetric matrix

, and similarly for

skew-symmetric

forms.

Let

ℬ

1

′

and

ℬ

2

′

be new bases, and

P

and

Q

the corresponding

change of basis matrices

. Then the new matrix is

C

′

=

P

T

⁢

C

⁢

Q

.

Title

matrix representation

of a bilinear form

Canonical name

MatrixRepresentationOfABilinearForm

Date of creation

2013-03-22 14:56:22

Last modified on

2013-03-22 14:56:22

Owner

vitriol (148)

Last modified by

vitriol (148)

Numerical id

5

Author

vitriol (148)

Entry type

Definition

Classification

msc 15A63

Classification

msc 11E39

Classification

msc 47A07

Generated on Fri Feb 9 18:30:08 2018 by

LaTeXML
non-degenerate quadratic form

non-degenerate quadratic form

Let

k

be a field of

characteristic

not 2. Then a

quadratic form

Q

over a

vector space

V

(over a field

k

) is said to be , if its associated

bilinear form

:

B

⁢

(

x

,

y

)

=

1

2

⁢

(

Q

⁢

(

x

+

y

)

-

Q

⁢

(

x

)

-

Q

⁢

(

y

)

)

is

non-degenerate

.

If we fix a basis

𝒃

for

V

, then

Q

⁢

(

x

)

can be written as

Q

⁢

(

x

)

=

x

T

⁢

A

⁢

x

for some

symmetric matrix

A

over

k

. Then it’s not hard to see that

Q

is non-degenerate iff

A

is

non-singular

. Because of this, a

non-degenerate quadratic form

is also known as a

non-singular

quadratic form. A third name for a non-degenerate quadratic form is that of a

regular quadratic form

.

Title

non-degenerate quadratic form

Canonical name

NondegenerateQuadraticForm

Date of creation

2013-03-22 15:05:58

Last modified on

2013-03-22 15:05:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A63

Classification

msc 11E39

Classification

msc 47A07

Synonym

non degenerate quadratic form

Synonym

non

singular

quadratic form

Defines

non-degenerate quadratic form

Defines

non-singular quadratic form

Defines

regular quadratic form

Generated on Fri Feb 9 18:30:10 2018 by

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orthogonal direct sum

orthogonal direct sum

Let

(

V

1

,

B

1

)

and

(

V

2

,

B

2

)

be two

vector spaces

, each equipped with a

symmetric bilinear form

. Form the

direct sum

of the two vector spaces

V

:=

V

1

⊕

V

2

. Next define a symmetric bilinear form

B

on

V

by

B

⁢

(

(

u

1

,

u

2

)

,

(

v

1

,

v

2

)

)

:=

B

1

⁢

(

u

1

,

v

1

)

+

B

2

⁢

(

u

2

,

v

2

)

,

where

u

1

,

v

1

∈

V

1

and

u

2

,

v

2

∈

V

2

. Since

B

⁢

(

(

u

1

,

0

)

,

(

u

2

,

0

)

)

=

B

1

⁢

(

u

1

,

u

2

)

, we see that

B

=

B

1

when the domain of

B

is restricted to

V

1

. Therefore,

V

1

can be viewed as a

subspace

of

V

with respect to

B

. The same holds for

V

2

.

Now suppose

(

u

,

0

)

∈

V

1

and

(

0

,

v

)

∈

V

2

are two arbitrary vectors. Then

B

⁢

(

(

u

,

0

)

,

(

0

,

v

)

)

=

B

1

⁢

(

u

,

0

)

+

B

2

⁢

(

0

,

v

)

=

0

+

0

=

0

. In other words,

V

1

and

V

2

are “

orthogonal

” to one another with respect to

B

.

From the above discussion, we say that

(

V

,

B

)

is the

orthogonal direct sum

of

(

V

1

,

B

1

)

and

(

V

2

,

B

2

)

. Clearly the above construction is unique (up to

linear isomorphisms

respecting the

bilinear forms

). As vectors from

V

1

and

V

2

can be seen as being “

perpendicular

” to each other, we appropriately write

V

as

V

1

⊥

V

2

.

Orthogonal Direct Sums of

Quadratic Spaces

.

Since a

symmetric

biliner form induces a

quadratic form

(on the same space), we can speak of orthogonal direct sums of quadratic spaces. If

(

V

1

,

Q

1

)

and

(

V

2

,

Q

2

)

are two quadratic spaces, then the orthogonal direct sum of

V

1

and

V

2

is the direct sum of

V

1

and

V

2

with the corresponding quadratic form defined by

Q

⁢

(

(

u

,

v

)

)

:=

Q

1

⁢

(

u

)

+

Q

2

⁢

(

v

)

.

It may be shown that any

n

-dimensional quadratic space

(

V

,

Q

)

is an orthogonal direct sum of

n

one-dimensional quadratic subspaces. The quadratic form associated with a one-dimensional quadratic space is nothing more than

a

⁢

x

2

(the form is uniquely determined by the single coefficient

a

), and the space associated with this form is generally written as

⟨

a

⟩

. A finite dimensional quadratic space

V

is commonly written as

⟨

a

1

⟩

⊥

⋯

⊥

⟨

a

n

⟩

,

or simply

⁢

⟨

a

1

,

…

,

a

n

⟩

,

where

n

is the

dimension

of

V

.

Remark.

The orthogonal direct sum may also be defined for vector spaces associated with bilinear forms that are

alternating

(

http://planetmath.org/AlternatingForm

),

skew symmetric

or

Hermitian

. The construction is

similar

to the one discussed above.

Title

orthogonal direct sum

Canonical name

OrthogonalDirectSum

Date of creation

2013-03-22 15:42:02

Last modified on

2013-03-22 15:42:02

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A63

Synonym

orthogonal sum

Generated on Fri Feb 9 18:30:13 2018 by

LaTeXML
perfect bilinear form

perfect bilinear form

Let

A

,

B

, and

C

be

abelian groups

. A

bilinear form

φ

:

A

×

B

→

C

is said to be

if the associated

group homomorphisms

A

→

Hom

⁡

(

B

,

C

)

a

↦

φ

⁢

(

a

,

⋅

)

and

B

→

Hom

⁡

(

A

,

C

)

b

↦

φ

⁢

(

⋅

,

b

)

are injective.

In particular, if

C

is finite then the finiteness of either

A

or

B

implies the finiteness of the other.

Title

perfect bilinear form

Canonical name

PerfectBilinearForm

Date of creation

2013-03-22 15:35:04

Last modified on

2013-03-22 15:35:04

Owner

matsuura (2984)

Last modified by

matsuura (2984)

Numerical id

5

Author

matsuura (2984)

Entry type

Definition

Classification

msc 15A63

Classification

msc 11E39

Generated on Fri Feb 9 18:30:15 2018 by

LaTeXML