v2010.10.26 - Convex Optimization

86 CHAPTER 2. CONVEX GEOMETRY2.5.0.0.1 Exercise. Subspace algebra.GivenR(A) + N(A T ) = R(B) + N(B T ) = R m (145)proveR(A) ⊇ N(B T ) ⇔ N(A T ) ⊆ R(B) (146)R(A) ⊇ R(B) ⇔ N(A T ) ⊆ N(B T ) (147)e.g., Theorem A.3.1.0.6.2.5.1 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 (148)and vector b∈R m ,a T mA {x∈ R n | Ax=b} =m⋂ { }x | aTi x=b ii=1(149)a halfspace-description. (113)For example: The intersection of any two independent 2.25 hyperplanesin R 3 is a line, whereas three independent hyperplanes intersect at apoint. In R 4 , the intersection of two independent hyperplanes is a plane2.25 Hyperplanes are said to be independent iff the normals defining them are linearlyindependent.