v2010.10.26 - Convex Optimization

2.5. SUBSPACE REPRESENTATIONS 852.5 Subspace representationsThere are two common forms of expression for Euclidean subspaces, bothcoming from elementary linear algebra: range form R and nullspace form N ;a.k.a, vertex-description and halfspace-description respectively.The fundamental vector subspaces associated with a matrix A∈ R m×n[331,3.1] are ordinarily related by orthogonal complementand of dimension:R(A T ) ⊥ N(A), N(A T ) ⊥ R(A) (137)R(A T ) ⊕ N(A) = R n , N(A T ) ⊕ R(A) = R m (138)dim R(A T ) = dim R(A) = rankA ≤ min{m,n} (139)with complementarity (a.k.a, conservation of dimension)dim N(A) = n − rankA , dim N(A T ) = m − rankA (140)These equations (137)-(140) comprise the fundamental theorem of linearalgebra. [331, p.95, p.138]From these four fundamental subspaces, the rowspace and range identifyone form of subspace description (range form or vertex-description (2.3.4))R(A T ) spanA T = {A T y | y ∈ R m } = {x∈ R n | A T y=x , y ∈R(A)} (141)R(A) spanA = {Ax | x∈ R n } = {y ∈ R m | Ax=y , x∈R(A T )} (142)while the nullspaces identify the second common form (nullspace form orhalfspace-description (113))N(A) {x∈ R n | Ax=0} = {x∈ R n | x ⊥ R(A T )} (143)N(A T ) {y ∈ R m | A T y=0} = {y ∈ R m | y ⊥ R(A)} (144)Range forms (141) (142) are realized as the respective span of the columnvectors in matrices A T and A , whereas nullspace form (143) or (144) is thesolution set to a linear equation similar to hyperplane definition (114). Yetbecause matrix A generally has multiple rows, halfspace-description N(A) isactually the intersection of as many hyperplanes through the origin; for (143),each row of A is normal to a hyperplane while each row of A T is a normalfor (144).