projective basis

projective basis

In the parent entry, we see how one may define

dimension

of a projective space inductively, from its

subspaces

starting with a point, then a line, and working its way up. Another way to define dimension start with defining dimensions of the

empty set

, a point, a line, and a plane to be

-

1

,

0

,

1

, and

2

, and then use the fact that any other projective space is

isomorphic

to the projective space

P

⁢

(

V

)

associated with a

vector space

V

, and then define the dimension to be the dimension of

V

, minus

1

. In this entry, we introduce a more natural way of defining dimensions, via the

concept

of a basis.

Throughout the discussion,

𝐏

is a projective space (as in any model satisfying the axioms of

projective geometry

).

Given a subset

S

of

𝐏

, the

span

of

S

, written

⟨

S

⟩

, is the smallest subspace of

𝐏

containing

S

. In other words,

⟨

S

⟩

is the

intersection

of all subspaces of

𝐏

containing

S

. Thus, if

S

is itself a subspace of

𝐏

,

⟨

S

⟩

=

S

. We also say that

S

spans

⟨

S

⟩

.

One may think of

⟨

⋅

⟩

as an

operation

on the

powerset

of

𝐏

. It is easy to verify that this operation is a

closure operator

. In

addition

,

⟨

⋅

⟩

is

algebraic

, in the sense that any point in

⟨

S

⟩

is in the span of a finite subset of

S

. In other words,

⟨

S

⟩

=

{

P

∣

P

∈

⟨

F

⟩

⁢

for some finite

⁢

F

⊆

S

}

.

Another

property

of

⟨

⋅

⟩

is the exchange property: for any subspace

U

, if

P

∉

U

, then for any point

Q

,

⟨

U

∪

{

P

}

⟩

=

⟨

U

∪

{

Q

}

⟩

iff

Q

∈

⟨

U

∪

{

P

}

⟩

-

U

.

A subset

S

of

𝐏

is said to be

projectively independent

, or simply

independent

, if, for any

proper subset

S

′

of

S

, the span of

S

′

is a proper subset of the span of

S

:

⟨

S

′

⟩

⊂

⟨

S

⟩

. This is the same as saying that

S

is a

minimal

spanning set

for

⟨

S

⟩

, in the sense that no proper subset of

S

spans

⟨

S

⟩

. Equivalently,

S

is independent iff for any

x

∈

S

,

⟨

S

-

{

x

}

⟩

≠

⟨

S

⟩

.

S

is called a

projective basis

, or simply

basis

for

𝐏

, if

S

is independent and spans

𝐏

.

All of the properties about spanning sets,

independent sets

, and bases for vector spaces have their projective counterparts. We list some of them here:

1.

Every projective space has a basis.

2.

If

S

1

,

S

2

are independent, then

⟨

S

1

∩

S

2

⟩

=

⟨

S

1

⟩

∩

⟨

S

2

⟩

.

3.

If

S

is independent and

P

∈

⟨

S

⟩

, then there is

Q

∈

S

such that

(

{

P

}

∪

S

)

-

{

Q

}

spans

⟨

S

⟩

.

4.

Let

B

be a basis for

𝐏

. If

S

spans

𝐏

, then

|

B

|

≤

|

S

|

. If

S

is independent, then

|

S

|

≤

|

B

|

. As a result, all bases for

𝐏

have the same

cardinality

.

5.

Every independent subset in

𝐏

may be extended to a basis for

𝐏

.

6.

Every spanning set for

𝐏

may be reduced to a basis for

𝐏

.

In light of items 1 and 4 above, we may define the

dimension

of

𝐏

to be the cardinality of its basis.

One of the main result on dimension is the dimension

formula

: if

U

,

V

are subspaces of

𝐏

, then

dim

⁡

(

U

)

+

dim

⁡

(

V

)

=

dim

⁡

(

U

∪

V

)

+

dim

⁡

(

U

∩

V

)

,

which is the counterpart of the same formula for vector subspaces of a vector space (see this entry (

http://planetmath.org/DimensionFormulaeForVectorSpaces

)).

References

1

A. Beutelspacher, U. Rosenbaum

Projective Geometry, From

Foundations

to Applications

, Cambridge University Press (2000)

Title

projective basis

Canonical

name

ProjectiveBasis

Date of creation

2013-03-22 19:14:38

Last modified on

2013-03-22 19:14:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05B35

Classification

msc 06C10

Classification

msc 51A05

Synonym

independent

Defines

span

Defines

projective independence

Defines

projectively independent

Defines

basis

Generated on Thu Feb 8 20:39:57 2018 by

LaTeXML