v2007.11.26 - Convex Optimization

74 CHAPTER 2. CONVEX GEOMETRY2.5.1 Subspace or affine subset ...Any particular vector subspace R p can be described as N(A) the nullspaceof some matrix A or as R(B) the range of some matrix B .More generally, we have the choice of expressing an n − m-dimensionalaffine subset in R n as the intersection of m hyperplanes, or as the offset spanof n − m vectors:2.5.1.1 ...as hyperplane intersectionAny affine subset A of dimension n−m can be described as an intersectionof m hyperplanes in R n ; given fat (m≤n) full-rank (rank = min{m , n})matrix⎡a T1 ⎤A =∆ ⎣ . ⎦∈ R m×n (124)and vector b∈R m ,a T mA ∆ = {x∈ R n | Ax=b} =m⋂ { }x | aTi x=b ii=1(125)a halfspace-description. (94)For example: The intersection of any two independent 2.22 hyperplanesin R 3 is a line, whereas three independent hyperplanes intersect at apoint. In R 4 , the intersection of two independent hyperplanes is a plane(Example 2.5.1.2.1), whereas three hyperplanes intersect at a line, four at apoint, and so on.For n>kA ∩ R k = {x∈ R n | Ax=b} ∩ R k =m⋂ { }x∈ R k | a i (1:k) T x=b ii=1(126)The result in2.4.2.2 is extensible; id est, any affine subset A also has avertex-description:2.22 Hyperplanes are said to be independent iff the normals defining them are linearlyindependent.