From Wikipedia, the free encyclopedia
Smallest affine subspace that contains a subset
In
mathematics , the affine hull or affine span of a
set S in
Euclidean space R n is the smallest
affine set containing S ,
[1] or equivalently, the
intersection of all affine sets containing S . Here, an affine set may be defined as the
translation of a
vector subspace .
The affine hull aff(S ) of S is the set of all
affine combinations of elements of S , that is,
aff
(
S
)
=
{
∑
i
=
1
k
α
i
x
i
|
k
>
0
,
x
i
∈
S
,
α
i
∈
R
,
∑
i
=
1
k
α
i
=
1
}
.
{\displaystyle \operatorname {aff} (S)=\left\{\sum _{i=1}^{k}\alpha _{i}x_{i}\,{\Bigg |}\,k>0,\,x_{i}\in S,\,\alpha _{i}\in \mathbb {R} ,\,\sum _{i=1}^{k}\alpha _{i}=1\right\}.}
Examples
The affine hull of the
empty set is the empty set.
The affine hull of a
singleton (a set made of one single element) is the singleton itself.
The affine hull of a set of two different points is the
line through them.
The affine hull of a set of three points not on one line is the
plane going through them.
The affine hull of a set of four points not in a plane in R 3 is the entire space R 3 .
Properties
For any subsets
S
,
T
⊆
X
{\displaystyle S,T\subseteq X}
aff
(
aff
S
)
=
aff
S
{\displaystyle \operatorname {aff} (\operatorname {aff} S)=\operatorname {aff} S}
aff
S
{\displaystyle \operatorname {aff} S}
is a
closed set if
X
{\displaystyle X}
is finite dimensional.
aff
(
S
+
T
)
=
aff
S
+
aff
T
{\displaystyle \operatorname {aff} (S+T)=\operatorname {aff} S+\operatorname {aff} T}
If
0
∈
S
{\displaystyle 0\in S}
then
aff
S
=
span
S
{\displaystyle \operatorname {aff} S=\operatorname {span} S}
.
If
s
0
∈
S
{\displaystyle s_{0}\in S}
then
aff
(
S
)
−
s
0
=
span
(
S
−
s
0
)
{\displaystyle \operatorname {aff} (S)-s_{0}=\operatorname {span} (S-s_{0})}
is a linear subspace of
X
{\displaystyle X}
.
aff
(
S
−
S
)
=
span
(
S
−
S
)
{\displaystyle \operatorname {aff} (S-S)=\operatorname {span} (S-S)}
.
So in particular,
aff
(
S
−
S
)
{\displaystyle \operatorname {aff} (S-S)}
is always a vector subspace of
X
{\displaystyle X}
.
If
S
{\displaystyle S}
is
convex then
aff
(
S
−
S
)
=
⋃
λ
>
0
λ
(
S
−
S
)
{\displaystyle \operatorname {aff} (S-S)=\displaystyle \bigcup _{\lambda >0}\lambda (S-S)}
For every
s
0
∈
S
{\displaystyle s_{0}\in S}
,
aff
S
=
s
0
+
cone
(
S
−
S
)
{\displaystyle \operatorname {aff} S=s_{0}+\operatorname {cone} (S-S)}
where
cone
(
S
−
S
)
{\displaystyle \operatorname {cone} (S-S)}
is the smallest
cone containing
S
−
S
{\displaystyle S-S}
(here, a set
C
⊆
X
{\displaystyle C\subseteq X}
is a cone if
r
c
∈
C
{\displaystyle rc\in C}
for all
c
∈
C
{\displaystyle c\in C}
and all non-negative
r
≥
0
{\displaystyle r\geq 0}
).
Hence
cone
(
S
−
S
)
{\displaystyle \operatorname {cone} (S-S)}
is always a linear subspace of
X
{\displaystyle X}
parallel to
aff
S
{\displaystyle \operatorname {aff} S}
.
Related sets
If instead of an affine combination one uses a
convex combination , that is, one requires in the formula above that all
α
i
{\displaystyle \alpha _{i}}
be non-negative, one obtains the
convex hull of S , which cannot be larger than the affine hull of S , as more restrictions are involved.
The notion of
conical combination gives rise to the notion of the
conical hull
If however one puts no restrictions at all on the numbers
α
i
{\displaystyle \alpha _{i}}
, instead of an affine combination one has a
linear combination , and the resulting set is the
linear span of S , which contains the affine hull of S .
References
Sources