v2009.01.01 - Convex Optimization

2.5. SUBSPACE REPRESENTATIONS 79
Range forms (131) (132) are realized as the respective span of the column
vectors in matrices A T and A , whereas nullspace form (133) or (134) is the
solution set to a linear equation similar to hyperplane definition (105). Yet
because matrix A generally has multiple rows, halfspace-description N(A) is
actually the intersection of as many hyperplanes through the origin; for (133),
each row of A is normal to a hyperplane while each row of A T is a normal
for (134).
2.5.1 Subspace or affine subset ...
Any particular vector subspace R p can be described as N(A) the nullspace
of some matrix A or as R(B) the range of some matrix B .
More generally, we have the choice of expressing an n − m-dimensional
affine subset in R n as the intersection of m hyperplanes, or as the offset span
of n − m vectors:
2.5.1.1 ...as hyperplane intersection
Any affine subset A of dimension n−m can be described as an intersection
of m hyperplanes in R n ; given fat (m≤n) full-rank (rank = min{m , n})
matrix
⎡
a T1 ⎤
A =
∆ ⎣ . ⎦∈ R m×n (135)
and vector b∈R m ,
a T m
A ∆ = {x∈ R n | Ax=b} =
m⋂ { }
x | a
T
i x=b i
i=1
(136)
a halfspace-description. (104)
For example: The intersection of any two independent 2.25 hyperplanes
in R 3 is a line, whereas three independent hyperplanes intersect at a
point. 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 a
point, and so on. A describes a subspace whenever b = 0 in (136).
2.25 Hyperplanes are said to be independent iff the normals defining them are linearly
independent.