v2010.10.26 - Convex Optimization

78 CHAPTER 2. CONVEX GEOMETRYA 1A 2A 30Figure 27: Any one particular point of three points illustrated does not belongto affine hull A i (i∈1, 2, 3, each drawn truncated) of points remaining.Three corresponding vectors in R 2 are, therefore, affinely independent (butneither linearly or conically independent).whereR(A) = {Ax | ∀x} (142)2.4.2.3 Affine independence, minimal setFor any particular affine set, a minimal set of points constituting itsvertex-description is an affinely independent generating set and vice versa.Arbitrary given points {x i ∈ R n , i=1... N} are affinely independent(a.i.) if and only if, over all ζ ∈ R N ζ T 1=1, ζ k = 0 (confer2.1.2)x i ζ i + · · · + x j ζ j − x k = 0, i≠ · · · ≠j ≠k = 1... N (123)has no solution ζ ; in words, iff no point from the given set can be expressedas an affine combination of those remaining. We deducel.i. ⇒ a.i. (124)Consequently, {x i , i=1... N} is an affinely independent set if and only if{x i −x 1 , i=2... N} is a linearly independent (l.i.) set. [[205,3] ] (Figure 27)XThis is equivalent to the property that the columns of1 T (for X ∈ R n×Nas in (76)) form a linearly independent set. [199,A.1.3]