Conformal Geometric Algebra in Stochastic Optimization Problems ...

2.3. EXTENDED CONCEPTS OF GA 69
Hence the transformation of the whole is actually the transformation of its parts.
What makes these properties so beneficial is, for instance, that equations or rules
may first be established in a simple coordinate system and do then also hold in their
reflected, rotated or translated versions. A case in point is the conformal geometric
algebra, which is the subject of chapter 3.
Make yourself aware of the possibility of versors acting on versors. Let nV n be
the reflection of the versor V in the unit vector n, n 2 = 1. The effect of the
transformation ensemble may easily be inferred writing
(nV n)x(n � V n) = n(V (nxn) � V )n.
So the action of V eventually takes place, but in a temporally different frame. It
amounts to the same, but nV n can as well be interpreted as the new modified
versor V ′ . Let, for instance V = v1v2 . ..vk, such that
� �� �
nV n = nv1v2 ...vkn = (nv1
1
n)(nv2n)...(nvkn) = v ′ 1v ′ 2 ...v ′ k = V ′ .
Last but not least, when applied to a k-blade, the versor likewise transforms the
dual of that blade
V A 〈k〉 � V = V (a1 ∧ a2 ∧ . .. ∧ ak) � V
= V (x1 ∧ x2 ∧ ... ∧ xn−k)I � V
(2.49)
= (−1) k(n−1) V (x1 ∧ x2 ∧ ... ∧ xn−k) � V I
= (−1) k(n−1) V A ∗ 〈k〉 � V I, (2.57)
where it was assumed that a1 ∧ a2 ∧ . .. ∧ ak = (x1 ∧ x2 ∧ ...xn−k)I.
2.3.5 Subspace Considerations
By the next couple of definitions a phrase like ‘... x lies in A 〈k〉 ’ can be stated
more precisely.
Definition 2.16 ( Outer product null space):
Given a blade A 〈k〉 ∈ �p,q, its outer product null space (OPNS) comprises all points
from � p,q that lie in the subspace represented by A 〈k〉 . The OPNS of A 〈k〉 , denoted
by �er(A 〈k〉 ), is defined as
�er(A 〈k〉 ) = {x ∈ � p,q |x ∧ A 〈k〉 = 0}.
Strictly speaking, the OPNS is supposed to be defined independently of the signature.
It should, however, be clear that x ∈ �er(A 〈k〉 ), with x = � n
i=1 xiei ∈ � p,q ,
is equivalent to x ′ ∈ �er(A 〈k〉 ), where x ′ = � n
i=1 xie ′ i ∈ �a,b , as long as a + b = n.
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