v2007.09.13 - Convex Optimization

2.1. CONVEX SET 35Figure 9: A slab is a convex Euclidean body infinite in extent butnot affine. Illustrated in R 2 , it may be constructed by intersectingtwo opposing halfspaces whose bounding hyperplanes are parallel but notcoincident. Because number of halfspaces used in its construction is finite,slab is a polyhedron. (Cartesian axes and vector inward-normal to eachhalfspace-boundary are drawn for reference.)2.1.2 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 (5)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.2.1.2.1 preservationLinear independence can be preserved under linear transformation. Givenmatrix Y ∆ = [y 1 y 2 · · · y N ]∈ R N×N , consider the mappingT(Γ) : R n×N → R n×N ∆ = ΓY (6)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 exists no nontrivial solution ζ ∈ R N toΓy 1 ζ i + · · · + Γy N−1 ζ N−1 + Γy N ζ N = 0 (7)By factoring Γ , we see that is ensured by linear independence of {y i ∈ R N }.