v2009.01.01 - Convex Optimization

72 CHAPTER 2. CONVEX GEOMETRY
A 1
A 2
A 3
0
Figure 23: Any one particular point of three points illustrated does not belong
to affine hull A i (i∈1, 2, 3, each drawn truncated) of points remaining.
Three corresponding vectors in R 2 are, therefore, affinely independent (but
neither linearly or conically independent).
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. [[179,3] ] (Figure 23)
X
This is equivalent to the property that the columns of
1 T (for X ∈ R n×N
as in (68)) form a linearly independent set. [173,A.1.3]
2.4.2.4 Preservation of affine independence
Independence in the linear (2.1.2.1), affine, and conic (2.10.1) senses can
be preserved under linear transformation. Suppose a matrix X ∈ R n×N (68)
holds an affinely independent set in its columns. Consider a transformation
T(X) : R n×N → R n×N ∆ = XY (116)
where the given matrix Y ∆ = [y 1 y 2 · · · y N ]∈ R N×N is represented by linear
operator T . Affine independence of {Xy i ∈ R n , i=1... N} demands (by
definition (114)) there exist no solution ζ ∈ R N ζ T 1=1, ζ k = 0, to
Xy i ζ i + · · · + Xy j ζ j − Xy k = 0, i≠ · · · ≠j ≠k = 1... N (117)