Page 449 - Advanced Linear Algebra

Affine Geometry 433

dim²? v @ ³ ~ dim²: b ;³ b
Finally, we have
dim²: b ;³ ~ dim²:³ b dim²;³ c dim²: q ;³

and Theorem 16.5 implies that
dim²? q @ ³ ~ dim²: q ;³

Affine Independence

We now discuss the affine counterpart of linear independence.

=
Theorem 16.7 Let ? be a nonempty set of vectors in . The following are
equivalent:
)
1 For all %? , the set
²? c %³ ± ¸ ¹
is linearly independent.
)
2 For all %? ,
%¤ affhull ²? ± ¸%¹³

)
3 For any vectors % ? ,

% ~ Á ~ ¬ ~ for all

)
4 For affine combinations of vectors in ,
?
% ~ % ¬ ~ for all

)
5 When ? ~ ¸% ÁÃÁ% ¹ is finite,

dim ² ²affhull ? ³ ³ ~ c
A set of vectors satisfying any any hence all of these conditions is said to be
(
)
?
affinely independent.
Proof. If 1) holds but there is an affine combination equal to ,
%

%~ %

~

where %£ % for all , then

²% c %³ ~

~