v2007.09.17 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 67A 1A 2A 30Figure 18: 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).2.4.2.4 Preservation of affine independenceIndependence in the linear (2.1.2.1), affine, and conic (2.10.1) senses canbe preserved under linear transformation. Suppose a matrix X ∈ R n×N (65)holds an affinely independent set in its columns. Consider a transformationT(X) : R n×N → R n×N ∆ = XY (105)where the given matrix Y ∆ = [y 1 y 2 · · · y N ]∈ R N×N is represented by linearoperator T . Affine independence of {Xy i ∈ R n , i=1... N} demands (bydefinition (103)) there exists no solution ζ ∈ R N ζ T 1=1, ζ k = 0, toXy i ζ i + · · · + Xy j ζ j − Xy k = 0, i≠ · · · ≠j ≠k = 1... N (106)By factoring X , we see that is ensured by affine independence of {y i ∈ R N }and by R(Y )∩ N(X) = 0 whereN(A) = {x | Ax=0} (121)