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430 Advanced Linear Algebra

b ~ ²% c%³ b ²% c%³ ~ % b % c² b ³%

%
for % ? . Since the sum of the coefficients of % , % and in the last

expression is , it follows that

b b% ~ % b % c² b c ³%  ?

:
and so b  ? c% ~ : . Thus, is a subspace of and ? ~ % b: is
=

a flat. The rest follows from Theorem 16.2.
Affine Hulls
The following definition is the analog of the subspace spanned by a collection
of vectors.
Definition Let be a nonempty set of vectors in .
?
=
1 The affine hull of , denoted by affhull ? ² ³ , is the smallest flat containing
)
?
?.
2 The affine span of ? , denoted by affspan ? ² ³ , is the set of all affine
)
?
combinations of vectors in .
Theorem 16.4 Let be a nonempty subset of . Then
?
=
affhull²?³ ~ affspan²?³ ~ % b span²? c %³
or equivalently, for a subspace of ,
=
:
% b :~ affspan ²?³ ¯ :~ span ²? c %³
Also,
dim ² ²affspan ? ³ ³ ~ dim span ² ? c % ³ ³
²
Proof. Theorem 16.3 implies that affspan²?³ ‹ affhull²?³ and so it is
sufficient to show that (~ affspan ²?³ is a flat, or equivalently, that for any
& (, the set : ~ ( c & is a subspace of . To this end, let
=

&~ %
Á
~
:
Then any two elements of have the form & c & and & c & , where

&~ % and &~ % 2 Á
Á
~ ~
(
are in . But if Á !  - , then