 Let
 , where
and are affine spaces.
Then T is said to be an affine transformation if:

T maps vectors to vectors and points to points

T is a linear transformation on the vectors

By Extension:
T preserves affine combinations on the points:
where
 Observations:

 Affine transformations map lines to lines:
 Affine transformations preserve ratios of distance
along a line
(converse is also true: preserves ratios
of such distances affine).
 Examples:
 rigid body motions (translations, rotations),
scales, shears, reflections.
 Affine vs Linear

 Theorem:
 Affine transformations map parallel lines to
parallel lines.
Proof: Let and .
Suppose , which implies
by linearity.
Then
and
and we see that the images of both lines are parallel.
 Suppose we only have T defined on points.

 Define
 as follows:
 There exists points Q and R such that .
 Define to be T(Q)T(R).
Note that Q and R are not unique.
The definition works for :
This can now be used to show that the definition is
well defined.
If QR=BC then
 How do we map points/vectors through an affine transformation?

 Let
 and be affine spaces.
 Question:
 What are the coordinates
of T(P) relative to ?
 Fact:
 An affine transformation is completely characterized
by the image of a frame in the domain:
If
then we can find by substitution and
gathering like terms. \