Page 445 - Advanced Linear Algebra
P. 445
Affine Geometry 429
under affine combinations of any two vectors in ? . These are not standard
terms.
The line containing two distinct vectors %Á & = is the set
%& ~ ¸ % b ² c ³& -¹ ~ & b º% c &»
of all affine combinations of and . Thus, a subset of is two-affine closed
?
=
%
&
if and only if ? contains the line through any two of its points.
Theorem 16.2 Let be a vector space over a field with char ² - ³ £ . Then a
=
-
subset of is affine closed if and only if it is two-affine closed.
?
=
Proof. The theorem is proved by induction on the number of terms in an
affine combination. The case ~ holds by assumption. Assume the result true
for affine combinations with fewer than terms and consider the affine
combination
' ~ % bÄb %
where . There are two cases to consider. If either of and is not equal
to , say £ , write
' ~ % b ² c ³ > % b Ä b % ?
c c
and if ~ ~ , then since char ²-³ £ 2, we may write
'~ > % b % ? b % b Ä b % 33
In either case, the inductive hypothesis applies to the expression inside the
square brackets and then to .
'
The requirement char²-³ £ is necessary, for if - ~ { , then the subset
? ~ ¸² Á ³Á ² Á ³Á ² Á ³¹
of - is two-affine closed but not affine closed. We can now characterize flats.
Theorem 16.3 A nonempty subset of a vector space is a flat if and only if
?
=
?
? ² is affine closed. Moreover, if char - ³ £ , then is a flat if and only if is
?
two-affine closed.
Proof. Let ?~ % b : be a flat and let % ~ % b ? , where : . If
' ~ , then
% ~ ²% b ³ ~ % b
% b :
and so ? is affine closed. Conversely, suppose that ? is affine closed, let
%? and let : ~? c %. If - and : then
