v2007.09.13 - Convex Optimization

2.5. SUBSPACE REPRESENTATIONS 752.5.1.2 ...as span of nullspace basisAlternatively, we may compute a basis for the nullspace of matrix A in (123)and then equivalently express the affine subset as its range plus an offset:DefineZ ∆ = basis N(A)∈ R n×n−m (126)so AZ = 0. Then we have the vertex-description,A = {x∈ R n | Ax = b} = { Zξ + x p | ξ ∈ R n−m} ⊆ R n (127)the offset span of n − m column vectors, where x p is any particular solutionto Ax = b .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 (124):{ ∣ [ ] }∣∣∣A ∆ 1 = x∈ R 4 a11 a 12 a 13 a 14x = ba 21 a 22 a 23 a 1 (128)24The nullspace of A 1 is two dimensional (from Z in (127)), 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 (129)44If the two planes are independent (meaning any line in one is linearlyindependent [ ] of any line from the other), they will intersect at a point becauseA1then is invertible;A 2A 1 ∩ A 2 ={ ∣ [ ] [ ]}∣∣∣x∈ R 4 A1 b1x =A 2 b 2(130)