affine combination
Definition
Let be a vector space over a division ring . An affine combination of a finite set of vectors is a linear combination of the vectors
such that subject to the condition . In effect, an affine combination is a weighted average of the vectors in question.
For example, is an affine combination of and provided that the characteristic of is not . is known as the midpoint of and . More generally, if does not divide , then
is an affine combination of the ’s. is the barycenter of .
Relations with Affine Subspaces
Assume now . Given , we can form the set of all affine combinations of the ’s. We have the following
is a finite dimensional affine subspace. Conversely, a finite dimensional affine subspace is the set of all affine combinations of a finite set of vectors in .
Proof.
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.
Remarks.
-
•
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.
Affine Independence
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:
-
1.
is affinely independent.
-
2.
every element in can be written as an affine combination of elements in in a unique fashion.
-
3.
for every , is linearly independent.
Proof.
We will proceed as follows: (1) implies (2) implies (3) implies (1).
(1) implies (2). If has two distinct representations , we may assume, say . So is invertible with inverse . Then
Furthermore,
So for any , we have
The sum of the coefficients is easily seen to be 1, which implies that is a spanning set of that is smaller than , a contradiction.
(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. ∎
Remarks.
-
•
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 .
Title | affine combination |
Canonical name | AffineCombination |
Date of creation | 2013-03-22 16:00:13 |
Last modified on | 2013-03-22 16:00:13 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 19 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 51A15 |
Synonym | affine independent |
Related topic | AffineGeometry |
Related topic | AffineTransformation |
Related topic | ConvexCombination |
Defines | affine independence |
Defines | affinely independent |
Defines | affine hull |