Page 444 - Advanced Linear Algebra
P. 444
428 Advanced Linear Algebra
Theorem 16.1 Let : and ; be subspaces of = and let ? ~ % b : and
@~ & b ; be flats in .
=
)
1 The following are equivalent:
)
a some translate of is in : b @ $ ? @ for some $ =
?
b some translate of is in : b ; # : ; for some # =
)
:
c ) : ;
2 The following are equivalent:
)
a ? ) and are translates: b $ ? ~ @ for some $ =
@
b : ) and are translates: b # : ~ ; for some # =
;
c ) :~ ;
3) ?q @ £ JÁ : ; ¯ ? @
4) ?q @ £ JÁ : ~ ; ¯ ? ~ @
5) If ? @ then ? @ @ ? or ? q @ ~J
,
)
6 ? @ if and only if some translation of one of these flats is contained in
the other.
Proof. If 1a) holds, then c& b$b%b: ; and so 1b) holds. If 1b) holds,
then #; and so : ~ ²# b :³ c #; and so 1c) holds. If 1c) holds, then
&c % b ? ~ &b : &b ; ~ @ and so 1a) holds. Part 2) is proved in a
similar manner.
For part 3), : ; implies that # b ? @ for some # = and so if
' ? q @ then # b ' @ and so # @ , which implies that ? @ .
Conversely, if ? @ then part 1) implies that : ; . Part 4) follows similarly.
We leave proof of 5) and 6) to the reader.
Affine Combinations
Let be a nonempty subset of . It is well known that
=
?
?
=
1) ? is a subspace of if and only if is closed under linear combinations,
or equivalently, is closed under linear combinations of any two vectors
?
in .
?
2) The smallest subspace of = containing ? is the set of all linear
combinations of elements of . In different language, the linear hull of ?
?
is equal to the linear span of .
?
We wish to establish the corresponding properties of affine subspaces of ,
=
beginning with the counterpart of a linear combination.
%
=
Definition Let be a vector space and let = . A linear combination
% bÄb %
where - and ~ is called an affine combination of the vectors . %
Let us refer to a nonempty subset of as affine closed if is closed under
?
?
=
any affine combination of vectors in ? and two-affine closed if ? is closed
