Page 452 - Advanced Linear Algebra
P. 452
436 Advanced Linear Algebra
Theorem 16.9 If char²-³ £ , then a function ¢ = ¦ = is an affine
transformation if and only if it preserves affine combinations of every pair of its
points, that is, if and only if
² % b ² c ³&³ ~ ²%³ b ² c ³ ²&³
Thus, if char²-³ £ , then a map is an affine transformation if and only if it
&
%
sends the line through and to the line through ²%³ and ²&³ . It is clear that
linear transformations are affine transformations. So are the following maps.
Definition Let #= . The affine map ; ¢ = ¦ = defined by
#
;²%³ ~ % b #
#
for all %= , is called translation by .
#
B
It is not hard to see that any composition ;k # , where ²= ³ , is affine.
Conversely, any affine map must have this form.
Theorem 16.10 A function ¢ = ¦ = is an affine transformation if and only if
it is a linear operator followed by a translation,
~ ; k
#
B
where #= and ²= . ³
Proof. We leave proof that ;k is an affine transformation to the reader. Let
#
be an affine map and suppose that ~ c' . Then ; k ~ . Moreover,
'
letting ~; k , we have
'
² "b #³ ~ ² " b #³ b'
~ ² " b # b ² c c ³ ³ b '
~ " b # c ² c c ³' b '
~ " b #
and so is linear.
Corollary 16.11
1 The composition of two affine transformations is an affine transformation.
)
)
2 An affine transformation ~ ; k # is bijective if and only if is bijective.
)
=
3 The set aff²= ³ of all bijective affine transformations on is a group under
composition of maps, called the affine group of .
=
Let us make a few group-theoretic remarks about aff²= ³ . The set trans²= ³ of all
translations of = is a subgroup of aff ² = ³ . We can define a function
B¢ aff
²=³ ¦ ²=³ by
# ²; k ³ ~
