Editing Geometric algebra (section)

=== Grade projection ===
A {{tmath|1= \Z }}-[[graded vector space]] structure can be established on a geometric algebra by use of the exterior product that is naturally induced by the geometric product.

Since the geometric product and the exterior product are equal on orthogonal vectors, this grading can be conveniently constructed by using an orthogonal basis {{tmath|1= \{e_1,\ldots,e_n\} }}.
 
Elements of the geometric algebra that are scalar multiples of <math>1</math> are of grade <math>0</math> and are called ''scalars''. Elements that are in the span of <math>\{e_1,\ldots,e_n\}</math> are of grade {{tmath|1= 1 }} and are the ordinary vectors. Elements in the span of <math>\{e_ie_j\mid 1\leq i<j\leq n\}</math> are of grade <math>2</math> and are the bivectors. This terminology continues through to the last grade of {{tmath|1= n }}-vectors. Alternatively, {{tmath|1= n }}-vectors are called [[pseudoscalar]]s,  {{tmath|1=(n-1)}}-vectors are called pseudovectors, etc. Many of the elements of the algebra are not graded by this scheme since they are sums of elements of differing grade. Such elements are said to be of ''mixed grade''. The grading of multivectors is independent of the basis chosen originally.

This is a grading as a vector space, but not as an algebra. Because the product of an {{tmath|1= r }}-blade and an {{tmath|1= s }}-blade is contained in the span of <math>0</math> through {{tmath|1= r+s }}-blades, the geometric algebra is a [[filtered algebra]].

A multivector <math>A</math> may be decomposed with the '''grade-projection operator''' {{tmath|1= \langle A \rangle _r }}, which outputs the grade-{{tmath|1= r }} portion of {{tmath|1= A }}. As a result:
: <math> A = \sum_{r=0}^{n} \langle A \rangle _r </math>

As an example, the geometric product of two vectors <math> a b = a \cdot b + a \wedge b = \langle a b \rangle_0 + \langle a b \rangle_2</math> since <math>\langle a b \rangle_0=a\cdot b</math> and <math>\langle a b \rangle_2 = a\wedge b</math> and {{tmath|1= \langle a b \rangle_i=0 }}, for <math>i</math> other than <math>0</math> and {{tmath|1= 2 }}.

A multivector <math>A</math> may also be decomposed into even and odd components, which may respectively be expressed as the sum of the even and the sum of the odd grade components above:
: <math> A^{[0]} = \langle A \rangle _0 + \langle A \rangle _2 + \langle A \rangle _4 + \cdots </math>
: <math> A^{[1]} = \langle A \rangle _1 + \langle A \rangle _3 + \langle A \rangle _5 + \cdots </math>

This is the result of forgetting structure from a {{tmath|1= \mathrm{Z} }}-[[graded vector space]] to {{tmath|1= \mathrm{Z}_2 }}-[[graded vector space]]. The geometric product respects this coarser grading. Thus in addition to being a {{tmath|1= \mathrm{Z}_2 }}-[[graded vector space]], the geometric algebra is a {{tmath|1= \mathrm{Z}_2 }}-[[graded algebra]], {{aka}} a [[superalgebra]].

Restricting to the even part, the product of two even elements is also even. This means that the even multivectors defines an ''[[even subalgebra]]''. The even subalgebra of an {{tmath|1= n }}-dimensional geometric algebra is [[algebra homomorphism|algebra-isomorphic]] (without preserving either filtration or grading) to a full geometric algebra of <math>(n-1)</math> dimensions. Examples include <math>\mathcal{G}^{[0]}(2,0) \cong \mathcal{G}(0,1)</math> and {{tmath|1= \mathcal{G}^{[0]}(1,3) \cong \mathcal{G}(3,0) }}.