Affine Combinations and Affine Dependence
An affine combination is a linear combination in which the sum of the coefficients is 1. Just as members of a set of vectors are linearly independent if none is a linear combination of the others, so also they are affinely independent if none is an affine combination of the others. The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace. For example, the affine span of a set of two points is the line that contains both; the affine span of a set of three non-collinear points is the plane that contains all three.
Vectors
- v1, v2, ..., vn
are linearly dependent if there exist scalars a1, a2, …,an, not all zero, for which
-
a1v1 + a2v2 + … + anvn = 0
(1)
Similarly they are affinely dependent if in addition the sum of coefficients is zero:
Read more about this topic: Affine Space
Famous quotes containing the words combinations and/or dependence:
“...black women write differently from white women. This is the most marked difference of all those combinations of black and white, male and female. It’s not so much that women write differently from men, but that black women write differently from white women. Black men don’t write very differently from white men.”
—Toni Morrison (b. 1931)
“All charming people have something to conceal, usually their total dependence on the appreciation of others.”
—Cyril Connolly (1903–1974)