v2009.01.01 - Convex Optimization

80 CHAPTER 2. CONVEX GEOMETRY
For n>k
A ∩ R k = {x∈ R n | Ax=b} ∩ R k =
m⋂ { }
x∈ R k | a i (1:k) T x=b i
i=1
(137)
The result in2.4.2.2 is extensible; id est, any affine subset A also has a
vertex-description:
2.5.1.2 ...as span of nullspace basis
Alternatively, we may compute a basis for nullspace of matrix A in (135)
and then equivalently express the affine subset A as its range plus an offset:
Define
Z ∆ = basis N(A)∈ R n×n−m (138)
so AZ = 0. Then we have the vertex-description in Z ,
A = {x∈ R n | Ax = b} = { Zξ + x p | ξ ∈ R n−m} ⊆ R n (139)
the offset span of n − m column vectors, where x p is any particular solution
to 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 vector
space. 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’s
resort to the tools acquired.
Suppose an intersection of two hyperplanes in four dimensions is specified
by a fat full-rank matrix A 1 ∈ R 2×4 (m = 2, n = 4) as in (136):
A 1 ∆ =
{ ∣ [ ] }
∣∣∣
x∈ R 4 a11 a 12 a 13 a 14
x = b
a 21 a 22 a 23 a 1
24
(140)
The nullspace of A 1 is two dimensional (from Z in (139)), so A 1 represents
a plane in four dimensions. Similarly define a second plane in terms of
A 2 ∈ R 2×4 :
{ ∣ [ ] }
∣∣∣
A ∆ 2 = x∈ R 4 a31 a 32 a 33 a 34
x = b
a 41 a 42 a 43 a 2 (141)
44