affine combination (original) (raw)

Definition

For example, v=12⁢v1+12⁢v2 is an affine combination of v1 and v2 provided that the characteristic of D is not 2. v is known as the midpoint of v1 and v2. More generally, if char⁡(D) does not divide m, then

is an affine combination of the vi’s. v is the barycenter of v1,…,vn.

Relations with Affine Subspaces

Assume now char⁡(D)=0. Given v1,…,vn∈V, we can form the set A of all affine combinations of the vi’s. We have the following

A is a finite dimensional affine subspace. Conversely, a finite dimensional affine subspace A is the set of all affine combinations of a finite set of vectors in A.

Proof.

Suppose A is the set of affine combinations of v1,…,vn. If n=1, then A is a singleton {v}, so A=0+v, where 0 is the null subspace of V. If n>1, we may pick a non-zero vector v∈A. Define S={a-v∣a∈A}. Then for any s∈S and d∈D, d⁢s=d⁢(a-v)=d⁢a+(1-d)⁢v-v. Since d⁢a+(1-d)⁢v∈A, d⁢s∈S. If s1,s2∈S, then 12⁢(s1+s2)=12⁢((a1-v)+(a2-v))=12⁢(a1+a2)-v∈S, since 12⁢(a1+a2)∈A. So 12⁢(s1+s2)∈S. Therefore, s1+s2=2⁢(12⁢(s1+s2))∈S. This shows that S is a vector subspace of V and that A=S+v is an affine subspace.

Conversely, let A be a finite dimensional affine subspace. Write A=S+v, where S is a subspace of V. Since dim⁡(S)=dim⁡(A)=n, S has a basis {s1,…,sn}. For each i=1,…,n, define vi=n⁢si+v. Given a∈A, we have

From this calculation, it is evident that a is an affine combination of v1,…,vn, and v. ∎

When A is the set of affine combinations of two distinct vectors v,w, we see that A is a line, in the sense that A=S+v, a translate of a one-dimensional subspace S (a line through 0). Every element in A has the form d⁢v+(1-d)⁢w, d∈D. 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:

A is an affine subspace iff for every pair of vectors in A, the line formed by the pair is also in A.

Note, however, that the A in the above corollary is not assumed to be finite dimensional.

Remarks.

• •
  If one of v1,…,vn is the zero vector, then A coincides with S. In other words, an affine subspace is a vector subspace if it contains the zero vector.
• •
  Given A={k1⁢v1+⋯+kn⁢vn∣vi∈V,ki∈D,∑ki=1}, the subset
  is also an affine subspace.

Affine Independence

Proof.

We will proceed as follows: (1) implies (2) implies (3) implies (1).

(2) implies (3). Pick v=v1. Suppose 0=s2⁢(v2-v1)+⋯+sn⁢(vn-v1). Expand and we have 0=(-s2-⋯-sn)⁢v1+s2⁢v2+⋯+sn⁢vn. So (1-s2-⋯-sn)⁢v1+s2⁢v2+⋯+sn⁢vn=v1∈A. By assumption, there is exactly one way to express v1, so we conclude that s2=⋯=sn=0.

(3) implies (1). If M were not minimal, then some v∈M could be expressed as an affine combination of the remaining vectors in M. So suppose v1=k2⁢v2+⋯+kn⁢vn. Since ∑ki=1, we can rewrite this as 0=k2⁢(v2-v1)+⋯+kn⁢(vn-v1). Since not all ki=0, N={v2-v1,…,vn-v1} is not linearly independent. ∎

Remarks.

• •
  If {v1,…,vn} is affinely independent set spanning A, then dim⁡(A)=n-1.
• •
  More generally, a set M (not necessarily finite) of vectors is said to be affinely independent if there is a vector v∈M, such that N={w-v∣v≠w∈M} is linearly independent (every finite subset of N 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 S of vectors, the affine hull of S is the smallest affine subspace A that contains every vector of S, denoted by Aff⁡(S). Every vector in Aff⁡(S) can be written as an affine combination of vectors in S.