Conformal Geometric Algebra in Stochastic Optimization Problems ...

2.1. AN AXIOMATIC DERIVATION 23
e1e123 = (e1e1)e23 = e23. The elements in � p,q and �p,q share at least one common
property - they square to ±1:
e 2 123 = (e1e2e3)(e1e2e3) = e1e2e3e1e2e3 = −e 2 1e 2 2e 2 3 ∈ {+1, −1}
In summary, the entire geometric algebra has been derived from the three axioms
(2.4), (2.5) and (2.12) in conjunction with the quadratic space � p,q . In some constructions
of geometric algebra it is additionally required to have a linear map
(inclusion map) i : � p,q −→ �p,q for the embedding of � p,q into �p,q. Here the
existence of i is implicitly assumed and not made explicit because � p,q can be considered
a linear subspace of �p,q. A vector a ∈ � p,q is therefore embedded into �p,q
the moment it gets involved in a calculation that makes its embedding necessary.
The usage of lower case letters for vectors is, in a manner of speaking, a notation
to indicate that the grade of the element equals one - higher order basis blades are
zero.
2.1.2 On and beyond the Products
Here the aim is to work with the currently known products in order to gain insights
into their functioning and about their meaning. At first, the outer product of two
vectors is examined in some detail.
Let a and b be two arbitrary but linearly independent vectors. By exploiting the
bilinearity of the geometric product it can be seen that
a ∧ (αa + βb)
b ∧ (αa + βb)
�
∝ a ∧ b α, β ∈ � \ {0}
is still, up to a scalar factor only, a ∧b. Hence moving the operands a and b within
the plane spanned by them leaves the outer product basically unchanged, which
is why a ∧ b reflects the linear subspace (plane) {p = αa + βb |α,β ∈ �}. Selfevidently,
this idea will be reinforced later on in section 2.2, when the necessary
concepts that are to be developed here will be available.
The subspace stays the same as long as the linear transformation of the constituent
vectors is regular (invertible). Consider, for instance, a transformation with matrix
A ∈ � 2×2
a ′ = (A11a + A21b)
b ′ = (A12a + A22b)
, with A =
According to equation (2.39), on page 46, it is
� A11 A12
A21 A22
a ′ ∧ b ′ = det(A)a ∧ b. (2.14)
Thus the outer product remains exactly unchanged iff the determinant of the transformation
matrix is one. The set of real k×k-matrices satisfying this criterion forms
�
.