v2010.10.26 - Convex Optimization

2.5. SUBSPACE REPRESENTATIONS 87(Example 2.5.1.2.1), whereas three hyperplanes intersect at a line, four at apoint, and so on. A describes a subspace whenever b = 0 in (149).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(150)The result in2.4.2.2 is extensible; id est, any affine subset A also has avertex-description:2.5.1.2 ...as span of nullspace basisAlternatively, we may compute a basis for nullspace of matrix A (E.3.1) andthen equivalently express affine subset A as its span plus an offset: DefineZ basis N(A)∈ R n×n−m (151)so AZ = 0. Then we have a vertex-description in Z ,A = {x∈ R n | Ax = b} = { Zξ + x p | ξ ∈ R n−m} ⊆ R n (152)the offset span of n − m column vectors, where x p is any particular solutionto Ax = b . For example, A describes a subspace whenever x p = 0.2.5.1.2.1 Example. Intersecting planes in 4-space.Two planes can intersect at a point in four-dimensional Euclidean vectorspace. It is easy to visualize intersection of two planes in three dimensions;a line can be formed. In four dimensions it is harder to visualize. So let’sresort to the tools acquired.Suppose an intersection of two hyperplanes in four dimensions is specifiedby a fat full-rank matrix A 1 ∈ R 2×4 (m = 2, n = 4) as in (149):{ ∣ [ ] }∣∣∣A 1 x∈ R 4 a11 a 12 a 13 a 14x = ba 21 a 22 a 23 a 1 (153)24The nullspace of A 1 is two dimensional (from Z in (152)), so A 1 representsa plane in four dimensions. Similarly define a second plane in terms ofA 2 ∈ R 2×4 :{ ∣ [ ] }∣∣∣A 2 x∈ R 4 a31 a 32 a 33 a 34x = ba 41 a 42 a 43 a 2 (154)44