Page 450 - Advanced Linear Algebra

434 Advanced Linear Algebra

Since is nonzero for some , this contradicts 1). Hence, 1) implies 2). Suppose

that 2) holds and
% ~

where ~ . If some , say , is nonzero then

%~ c ² ° ³%  affhull ²? ± ¸% ¹³

€
which contradicts 2) and so ~ for all . Hence, 2) implies 3).

If 3) holds and the affine combinations satisfy
% ~ %

then
² c ³% ~

and since ² c ³ ~ c ~ , it follows that ~ for all . Hence, 4)

holds. Thus, it is clear that 3) and 4) are equivalent. If 3) holds and
²% c %³ ~

, then
for %£%
4 % c % ~
5

and so 3) implies that ~ for all .

Finally, suppose that ? ~ ¸% ÁÃÁ% ¹ . Since

dim²affhull ²?³³ ~ dim²º? c % »³

it follows that 5) holds if and only if ²? c % ³ ± ¸ ¹ , which has size c , is

linearly independent.
Affinely independent sets enjoy some of the basic properties of linearly
independent sets. For example, a nonempty subset of an affinely independent set
is affinely independent. Also, any nonempty set ? contains an affinely
independent set.
Since the affine hull /~ affhull ²?³ of an affinely independent set is not the
?
affine hull of any proper subset of ? , we deduce that ? is a minimal affine
spanning set of its affine hull.