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:
“The more elevated a culture, the richer its language. The number of words and their combinations depends directly on a sum of conceptions and ideas; without the latter there can be no understandings, no definitions, and, as a result, no reason to enrich a language.”
—Anton Pavlovich Chekhov (1860–1904)
“This immediate dependence of language upon nature, this conversion of an outward phenomenon into a type of somewhat in human life, never loses its power to affect us. It is this which gives that piquancy to the conversation of a strong-natured farmer or backwoodsman, which all men relish.”
—Ralph Waldo Emerson (1803–1882)