v2010.10.26 - Convex Optimization

2.1. CONVEX SET 372.1.2 linear independenceArbitrary given vectors in Euclidean space {Γ i ∈ R n , i=1... N} are linearlyindependent (l.i.) if and only if, for all ζ ∈ R NΓ 1 ζ 1 + · · · + Γ N−1 ζ N−1 − Γ N ζ N = 0 (6)has only the trivial solution ζ = 0 ; in other words, iff no vector from thegiven set can be expressed as a linear combination of those remaining.Geometrically, two nontrivial vector subspaces are linearly independent iffthey intersect only at the origin.2.1.2.1 preservation of linear independence(confer2.4.2.4,2.10.1) Linear transformation preserves linear dependence.[227, p.86] Conversely, linear independence can be preserved under lineartransformation. Given Y = [y 1 y 2 · · · y N ]∈ R N×N , consider the mappingT(Γ) : R n×N → R n×N ΓY (7)whose domain is the set of all matrices Γ∈ R n×N holding a linearlyindependent set columnar. Linear independence of {Γy i ∈ R n , i=1... N}demands, by definition, there exist no nontrivial solution ζ ∈ R N toΓy 1 ζ i + · · · + Γy N−1 ζ N−1 − Γy N ζ N = 0 (8)By factoring out Γ , we see that triviality is ensured by linear independenceof {y i ∈ R N }.2.1.3 Orthant:name given to a closed convex set that is the higher-dimensionalgeneralization of quadrant from the classical Cartesian partition of R 2 ; aCartesian cone. The most common is the nonnegative orthant R n + or R n×n+(analogue to quadrant I) to which membership denotes nonnegative vectorormatrix-entries respectively; e.g.,R n + {x∈ R n | x i ≥ 0 ∀i} (9)The nonpositive orthant R n − or R n×n− (analogue to quadrant III) denotesnegative and 0 entries. Orthant convexity 2.4 is easily verified bydefinition (1).2.4 All orthants are selfdual simplicial cones. (2.13.5.1,2.12.3.1.1)