properties of an affine transformation

properties of an affine transformation

In this entry, we prove some of the basic

properties of affine transformations

. Let

α

:

A

1

→

A

2

be an

affine transformation

and

[

α

]

:

V

1

→

V

2

its associated

linear transformation

.

Proposition 1

.

α

is

one-to-one

iff

[

α

]

is.

Proof.

Next, suppose

α

is one-to-one, and

T

⁢

(

v

)

=

0

for some

v

∈

V

1

. Let

P

,

Q

∈

A

1

with

f

1

⁢

(

P

,

Q

)

=

v

. Then

0

=

[

α

]

⁢

(

v

)

=

[

α

]

⁢

(

f

1

⁢

(

P

,

Q

)

)

=

f

1

⁢

(

α

⁢

(

P

)

,

α

⁢

(

Q

)

)

, which implies that

α

⁢

(

P

)

=

α

⁢

(

Q

)

, and therefore

P

=

Q

by

assumption

.

Conversely

, suppose

[

α

]

is one-to-one, and

α

⁢

(

P

)

=

α

⁢

(

Q

)

. Then

[

α

]

⁢

(

f

1

⁢

(

P

,

Q

)

)

=

f

2

⁢

(

α

⁢

(

P

)

,

α

⁢

(

Q

)

)

=

0

, so that

f

1

⁢

(

P

,

Q

)

=

0

, and consequently

P

=

Q

, showing that

α

is one-to-one.
∎

Proposition 2

.

α

is onto iff

[

α

]

is.

Proof.

Suppose

α

is onto. Let

w

∈

V

2

, so there are

X

,

Y

∈

A

2

such that

f

2

⁢

(

X

,

Y

)

=

w

. Since

α

is onto, there are

P

,

Q

∈

A

1

with

α

⁢

(

P

)

=

X

and

α

⁢

(

Q

)

=

Y

. So

w

=

f

2

⁢

(

X

,

Y

)

=

f

2

⁢

(

α

⁢

(

P

)

,

α

⁢

(

Q

)

)

=

[

α

]

⁢

(

f

1

⁢

(

P

,

Q

)

)

. Hence

[

α

]

is onto. Conversely, assume

[

α

]

be onto, and pick

Y

∈

A

2

. Take an arbitrary point

P

∈

A

1

and set

X

=

α

⁢

(

P

)

. There is

v

∈

V

1

such that

[

α

]

⁢

(

v

)

=

f

2

⁢

(

X

,

Y

)

, since

[

α

]

is onto. Let

Q

∈

A

1

such that

f

1

⁢

(

P

,

Q

)

=

v

. Then

f

2

⁢

(

X

,

α

⁢

(

Q

)

)

=

f

2

⁢

(

α

⁢

(

P

)

,

α

⁢

(

Q

)

)

=

[

α

]

⁢

(

f

1

⁢

(

P

,

Q

)

)

=

[

α

]

⁢

(

v

)

=

f

2

⁢

(

X

,

Y

)

. But

f

2

⁢

(

X

,

-

)

is a

bijection

, we must have

Y

=

α

⁢

(

Q

)

, showing that

α

is onto.
∎

Corollary 1

.

α

is a bijection iff

[

α

]

is.

Proposition 3

.

A

bijective

affine transformation

α

:

A

1

→

A

2

is an affine isomorphism.

Proof.

Suppose an affine transformation

α

:

A

1

→

A

2

is a bijection. We want to show that

α

-

1

:

A

2

→

A

1

is an affine transformation. Pick any

X

,

Y

∈

A

2

, then

[

α

]

⁢

(

f

1

⁢

(

α

-

1

⁢

(

X

)

,

α

-

1

⁢

(

Y

)

)

)

=

f

2

⁢

(

X

,

Y

)

.

By the corollary above,

[

α

]

is bijective, and hence a

linear isomorphism

. So

f

1

⁢

(

α

-

1

⁢

(

X

)

,

α

-

1

⁢

(

Y

)

)

=

[

α

]

-

1

⁢

(

f

2

⁢

(

X

,

Y

)

)

.

This shows that

α

-

1

is an affine transformation whose assoicated linear transformation is

[

α

]

-

1

.
∎

Proposition 4

.

Two affine spaces associated with the same

vector space

V

are affinely isomorphic.

Proof.

In fact, all we need to do is to show that

(

A

,

f

)

is

isomorphic

to

(

V

,

g

)

, where

g

is given by

g

⁢

(

v

,

w

)

=

w

-

v

. Pick any

P

∈

A

, then

α

:=

f

⁢

(

P

,

-

)

:

A

→

V

is a bijection. For any

v

∈

V

, there is a unique

Q

∈

A

such that

v

=

f

⁢

(

P

,

Q

)

. Then

1

V

⁢

(

f

⁢

(

X

,

Y

)

)

=

f

⁢

(

X

,

Y

)

=

f

⁢

(

P

,

Y

)

-

f

⁢

(

P

,

X

)

=

α

⁢

(

Y

)

-

α

⁢

(

X

)

=

g

⁢

(

α

⁢

(

X

)

,

α

⁢

(

Y

)

)

, showing that

1

V

is the associated linear transformation of

α

.
∎

Proposition 5

.

Any affine transformation is a linear transformation between the corresponding induced vector spaces. In other words, if

α

:

A

→

B

is affine, then

α

:

A

P

→

B

α

⁢

(

P

)

is linear.

Proof.

Suppose

Q

,

R

,

S

∈

A

are such that

Q

+

R

=

S

, or

f

1

⁢

(

P

,

Q

)

+

f

1

⁢

(

P

,

R

)

=

f

1

⁢

(

P

,

S

)

. Then

f

2

⁢

(

α

⁢

(

P

)

,

α

⁢

(

S

)

)

=

[

α

]

⁢

(

f

1

⁢

(

P

,

S

)

)

=

[

α

]

⁢

(

f

1

⁢

(

P

,

Q

)

+

f

1

⁢

(

P

,

R

)

)

=

[

α

]

⁢

(

f

1

⁢

(

P

,

Q

)

)

+

[

α

]

⁢

(

f

1

⁢

(

P

,

R

)

)

=

f

2

⁢

(

α

⁢

(

P

)

,

α

⁢

(

Q

)

)

+

f

2

⁢

(

α

⁢

(

P

)

,

α

⁢

(

R

)

)

,

which is

equivalent

to

α

⁢

(

Q

)

+

α

⁢

(

R

)

=

α

⁢

(

S

)

=

α

⁢

(

Q

+

R

)

.

Next, suppose

d

⁢

Q

=

R

, or

d

⁢

f

1

⁢

(

P

,

Q

)

=

f

1

⁢

(

P

,

R

)

, where

d

∈

D

. Then

f

2

⁢

(

α

⁢

(

P

)

,

α

⁢

(

R

)

)

=

[

α

]

⁢

(

f

1

⁢

(

P

,

R

)

)

=

[

α

]

⁢

(

d

⁢

f

1

⁢

(

P

,

Q

)

)

=

d

⁢

[

α

]

⁢

(

f

1

⁢

(

P

,

Q

)

)

=

d

⁢

f

2

⁢

(

α

⁢

(

P

)

,

α

⁢

(

Q

)

)

,

which is equivalent to

d

⁢

α

⁢

(

Q

)

=

α

⁢

(

R

)

=

α

⁢

(

d

⁢

Q

)

.
∎

Proposition 6

.

If

(

V

,

f

)

is an affine space associated with the vector space

V

, then the

direction

f

is given by

f

⁢

(

v

,

w

)

=

T

⁢

(

w

-

v

)

for some linear isomorphism (

invertible linear transformation

)

T

.

Proof.

By

proposition

4,

(

V

,

f

)

is affinely isomorphic to

(

V

,

g

)

with

g

⁢

(

v

,

w

)

=

w

-

v

. Suppose

α

:

(

V

,

f

)

→

(

V

,

g

)

is the affine isomorphism. Then

[

α

]

⁢

(

f

⁢

(

v

,

w

)

)

=

g

⁢

(

α

⁢

(

v

)

,

α

⁢

(

w

)

)

=

α

⁢

(

w

)

-

α

⁢

(

v

)

. Since

[

α

]

is a linear isomorphism,

f

⁢

(

v

,

w

)

=

[

α

]

-

1

⁢

(

α

⁢

(

w

)

)

-

[

α

]

-

1

⁢

(

α

⁢

(

v

)

)

. By proposition 5,

α

itself is linear, so

f

⁢

(

v

,

w

)

=

(

[

α

]

-

1

∘

α

)

⁢

(

w

-

v

)

. Set

T

=

[

α

]

-

1

∘

α

. Then

T

is linear and

invertible

since

α

is, our assertion is proved.
∎

Title

properties of an affine transformation

Canonical name

PropertiesOfAnAffineTransformation

Date of creation

2013-03-22 18:31:53

Last modified on

2013-03-22 18:31:53

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51A10

Classification

msc 15A04

Classification

msc 51A15

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