Editing Geometric algebra (section)

=== Unit pseudoscalars ===
Unit pseudoscalars are blades that play important roles in GA. A '''unit pseudoscalar''' for a non-degenerate subspace <math>W</math> of <math>V</math> is a blade that is the product of the members of an orthonormal basis for {{tmath|1= W }}. It can be shown that if <math>I</math> and <math>I'</math> are both unit pseudoscalars for {{tmath|1= W }}, then <math>I = \pm I'</math> and {{tmath|1= I^2 = \pm 1 }}. If one doesn't choose an orthonormal basis for {{tmath|1= W }}, then the [[Plücker embedding]] gives a vector in the exterior algebra but only up to scaling. Using the vector space isomorphism between the geometric algebra and exterior algebra, this gives the equivalence class of <math>\alpha I</math> for all {{tmath|1= \alpha \neq 0 }}. Orthonormality gets rid of this ambiguity except for the signs above.

Suppose the geometric algebra <math>\mathcal{G}(n,0)</math> with the familiar positive definite inner product on <math>\R^n</math> is formed. Given a plane (two-dimensional subspace) of&nbsp;{{tmath|1= \R^n }}, one can find an orthonormal basis <math>\{ b_1, b_2 \}</math> spanning the plane, and thus find a unit pseudoscalar <math>I = b_1 b_2</math> representing this plane. The geometric product of any two vectors in the span of <math>b_1</math> and <math>b_2</math> lies in {{tmath|1= \{ \alpha_0 + \alpha_1 I \mid \alpha_i \in \R \} }}, that is, it is the sum of a {{tmath|1= 0 }}-vector and a {{tmath|1= 2 }}-vector.

By the properties of the geometric product, {{tmath|1= I^2 = b_1 b_2 b_1 b_2 = -b_1 b_2 b_2 b_1 = -1 }}. The resemblance to the [[imaginary unit]] is not incidental: the subspace <math> \{ \alpha_0 + \alpha_1 I \mid \alpha_i \in \R \} </math> is {{tmath|1= \R }}-algebra isomorphic to the [[complex number]]s. In this way, a copy of the complex numbers is embedded in the geometric algebra for each two-dimensional subspace of <math>V</math> on which the quadratic form is definite.

It is sometimes possible to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in the real algebra that square to {{tmath|1= -1 }}, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces.

In {{tmath|1= \mathcal{G}(3,0) }}, a further familiar case occurs. Given a standard basis consisting of orthonormal vectors <math>e_i</math> of {{tmath|1= V }}, the set of ''all'' {{tmath|1= 2 }}-vectors is spanned by
: <math> \{ e_3 e_2 , e_1 e_3 , e_2 e_1 \} .</math>
Labelling these {{tmath|1= i }}, <math>j</math> and <math>k</math> (momentarily deviating from our uppercase convention), the subspace generated by {{tmath|1= 0 }}-vectors and {{tmath|1= 2 }}-vectors is exactly {{tmath|1=  \{ \alpha_0 + i \alpha_1 + j \alpha_2 + k \alpha_3 \mid \alpha_i \in \R\} }}. This set is seen to be the even subalgebra of {{tmath|1= \mathcal{G}(3,0) }}, and furthermore is isomorphic as an {{tmath|1= \R }}-algebra to the [[quaternion]]s, another important algebraic system.