Page 451 - Advanced Linear Algebra

Affine Geometry 435

Affine Bases and Barycentric Coordinates
We have seen that a set is affinely independent if and only if the set
?
? ~ ²? c %³ ± ¸ ¹
%
is linearly independent. We have also seen that for a subsapce of ,
:
=
% b :~ affspan ²?³ ¯ :~ span ²? ³
%
8
Therefore, if by analogy, we define a subset of a flat (~ % b : to be an
affine basis for if is affinely independent and affspan ² 8 ³ ~ ( , then is an
8
(
8
affine basis for %b: if and only if 8 % is a basis for .
:

8
Theorem 16.8 Let (~ % b : be a flat of dimension . Let ~ ²% Á Ã Á % ³ be

an ordered basis for and let ² b %³ r ¸%¹ ~ ²% b %Á Ã Á % b %Á %³ be an
8
:

ordered affine basis for . Then every #  ( has a unique expression as an
(
affine combination
%

# ~ % bÄb % b b

The coefficients are called the barycentric coordinates of with respect to

#
the ordered affine basis 8 b% .
For example, in s , a plane is a flat of the form (~ % b º# Á # » where

8 s ~ ²#Á #³ is an ordered basis for a two-dimensional subspace of . Then
8
² b %³ r ¸%¹ ~ ²# b %Á # b %Á %³ ~ ² Á Á ³

are barycentric coordinates for the plane, that is, any #( has the form
b b

where b b ~ .

Affine Transformations
Now let us discuss some properties of maps that preserve affine structure.
Definition A function ¢ = ¦ = that preserves affine combinations, that is, for
which
~ ¬ 5 ~ 4 ²% ³
%

(
is called an affine transformation or affine map , or affinity) .
We should mention that some authors require that be bijective in order to be

an affine map. The following theorem is the analog of Theorem 16.2.