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:
“You should try to understand every thing you see and hear; to act and judge for yourselves; to remember you each have a soul of your own to account for; M a mind of your own to improve. When you once get these ideas fixed, and learn to act upon them, no man or set of men, no laws, customs, or combinations of them can seriously oppress you.”
—Jane Grey Swisshelm (1815–1884)
“As, therefore, we can have no dependence upon morality without religion;Mso, on the other hand, there is nothing better to be expected from religion without morality;Mnevertheless, ‘tis no prodigy to see a man whose real moral character stands very low, who yet entertains the highest notion of himself, in the light of a religious man.”
—Laurence Sterne (1713–1768)