such that subject to the condition . In effect, an affine combination is a weighted average of the vectors in question.
Relations with Affine Subspaces
Assume now . Given , we can form the set of all affine combinations of the ’s. We have the following
Suppose is the set of affine combinations of . If , then is a singleton , so , where 0 is the null subspace of . If , we may pick a non-zero vector . Define . Then for any and , . Since , . If , then , since . So . Therefore, . This shows that is a vector subspace of and that is an affine subspace.
Conversely, let be a finite dimensional affine subspace. Write , where is a subspace of . Since , has a basis . For each , define . Given , we have
From this calculation, it is evident that is an affine combination of , and . ∎
When is the set of affine combinations of two distinct vectors , we see that is a line, in the sense that , a translate of a one-dimensional subspace (a line through 0). Every element in has the form , . Inspecting the first part of the proof in the previous proposition, we see that the argument involves no more than two vectors at a time, so the following useful corollary is apparant:
is an affine subspace iff for every pair of vectors in , the line formed by the pair is also in .
Note, however, that the in the above corollary is not assumed to be finite dimensional.
If one of is the zero vector, then coincides with . In other words, an affine subspace is a vector subspace if it contains the zero vector.
Given , the subset
is also an affine subspace.
Since every element in a finite dimensional affine subspace is an affine combination of a finite set of vectors in , we have the similar concept of a spanning set of an affine subspace. A minimal spanning set of an affine subspace is said to be affinely independent. We have the following three equivalent characterization of an affinely independent subset of a finite dimensional affine subspace:
is affinely independent.
every element in can be written as an affine combination of elements in in a unique fashion.
for every , is linearly independent.
We will proceed as follows: (1) implies (2) implies (3) implies (1).
So for any , we have
(2) implies (3). Pick . Suppose . Expand and we have . So . By assumption, there is exactly one way to express , so we conclude that .
(3) implies (1). If were not minimal, then some could be expressed as an affine combination of the remaining vectors in . So suppose . Since , we can rewrite this as . Since not all , is not linearly independent. ∎
If is affinely independent set spanning , then .
More generally, a set (not necessarily finite) of vectors is said to be affinely independent if there is a vector , such that is linearly independent (every finite subset of is linearly independent). The above three characterizations are still valid in this general setting. However, one must be careful that an affine combination is a finitary operation so that when we take the sum of an infinite number of vectors, we have to realize that only a finite number of them are non-zero.
Given any set of vectors, the affine hull of is the smallest affine subspace that contains every vector of , denoted by . Every vector in can be written as an affine combination of vectors in .
|Date of creation||2013-03-22 16:00:13|
|Last modified on||2013-03-22 16:00:13|
|Last modified by||CWoo (3771)|