v2009.01.01 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 73
C
H −
H +
0 > κ 3 > κ 2 > κ 1
{z ∈ R 2 | a T z = κ 1 }
a
{z ∈ R 2 | a T z = κ 2 }
{z ∈ R 2 | a T z = κ 3 }
Figure 24: Each shaded line segment {z ∈ C | a T z = κ i } belonging to set
C ⊂ R 2 shows intersection with hyperplane parametrized by scalar κ i ; each
shows a (linear) contour in vector z of equal inner product with normal
vector a . Cartesian axes drawn for reference. (confer Figure 63)
By factoring out X , we see that is ensured by affine independence of
{y i ∈ R N } and by R(Y )∩ N(X) = 0 where
N(A) = {x | Ax=0} (133)
2.4.2.5 affine maps
Affine transformations preserve affine hulls. Given any affine mapping T of
vector spaces and some arbitrary set C [266, p.8]
aff(T C) = T(aff C) (118)