Editing Geometric algebra (section)

== Definition and notation ==
There are a number of different ways to define a geometric algebra. Hestenes's original approach was axiomatic,{{sfn|ps=|Hestenes|Sobczyk|1984|p=3–5}} "full of geometric significance" and equivalent to the universal{{efn|A 'universal' algebra is the most "complete" or least degenerate algebra that satisfies all the defining equations. In this article, by 'Clifford algebra' we mean the universal Clifford algebra.}} Clifford algebra.{{sfn|ps=|Aragón|Aragón|Rodríguez|1997|p=101}}
Given a finite-dimensional vector space {{tmath|1= V }} over a [[Field (mathematics)|field]] {{tmath|1= F }} with a symmetric bilinear form (the ''inner product'',{{efn|name=inner|The term ''inner product'' as used in geometric algebra refers to the symmetric bilinear form on the {{tmath|1= 1 }}-vector subspace, and is a synonym for the ''scalar product'' of a [[pseudo-Euclidean vector space]], not the [[inner product]] on a normed vector space. Some authors may extend the meaning of ''inner product'' to the entire algebra, but there is little consensus on this. Even in texts on geometric algebras, the term is not universally used.}} e.g., the Euclidean or [[Lorentzian metric]]) {{tmath|1= g : V \times V \to F }}, the '''geometric algebra''' of the [[quadratic space]] {{tmath|1= (V, g) }} is the [[Clifford algebra]] {{tmath|1= \operatorname{Cl}(V, g) }}, an element of which is called a multivector. The Clifford algebra is commonly defined as a [[Quotient associative algebra|quotient algebra]] of the [[tensor algebra]], though this definition is abstract, so the following definition is presented without requiring [[abstract algebra]].

; Definition : A unital associative algebra {{tmath|1= \operatorname{Cl}(V, g) }} with a ''nondegenerate'' symmetric bilinear form {{tmath|1= g : V \times V \to F }} is the Clifford algebra of the quadratic space {{tmath|1= (V, g) }} if{{sfn|ps=|Lounesto|2001|p=190}} 
;* :
;* it contains {{tmath|1= F }} and {{tmath|1= V }} as distinct subspaces
;* {{tmath|1= a^2 = g(a,a)1 }} for {{tmath|1= a \in V }} 
;* {{tmath|1= V }} generates {{tmath|1= \operatorname{Cl}(V, g) }} as an algebra 
;* {{tmath|1= \operatorname{Cl}(V, g) }} is not generated by any proper subspace of {{tmath|1= V }}.

To cover degenerate symmetric bilinear forms, the last condition must be modified.{{efn|It may be replaced by the condition that{{sfn|ps=|Lounesto|2001|p=191}} the product of any set of linearly independent vectors in {{tmath|1= V }} must not be in {{tmath|1= F }} or that{{sfn|ps=|Vaz|da Rocha|2016|p=58|loc=Theorem 3.1}} the dimension of the algebra must be {{tmath|1= 2^{\dim V} }}.}} It can be shown that these conditions uniquely characterize the geometric product.

For the remainder of this article, only the [[Real number|real]] case, {{tmath|1= F = \R }}, will be considered. The notation {{tmath| 1=\mathcal{G}(p,q) }} (respectively {{tmath|1= \mathcal{G}(p,q,r) }}) will be used to denote a geometric algebra for which the bilinear form {{tmath|1= g }} has the [[Metric signature|signature]] {{tmath|1= (p,q) }} (respectively {{tmath|1= (p,q,r) }}).

The product in the algebra is called the ''geometric product'', and the product in the contained exterior algebra is called the ''exterior product'' (frequently called the ''wedge product'' or the ''outer product''{{efn|The term ''outer product'' used in geometric algebra conflicts with the meaning of ''[[outer product]]'' elsewhere in mathematics}}). It is standard to denote these respectively by juxtaposition (i.e., suppressing any explicit multiplication symbol) and the symbol {{tmath|1= \wedge }}.

The above definition of the geometric algebra is still somewhat abstract, so we summarize the properties of the geometric product here. For multivectors {{tmath|1= A, B, C\in \mathcal{G}(p,q) }}:
* {{tmath|1= AB \in \mathcal{G}(p,q) }} ([[Closure (mathematics)|closure]])
* {{tmath|1= 1A = A1 = A }}, where {{tmath|1= 1 }} is the identity element (existence of an [[identity element]])
* {{tmath|1= A(BC)=(AB)C }} ([[associativity]])
* {{tmath|1= A(B+C)=AB+AC }} and {{tmath|1= (B+C)A=BA+CA }} ([[distributivity]])
* {{tmath|1= a^2 = g(a,a)1 }} for {{tmath|1= a \in V }}.

The exterior product has the same properties, except that the last property above is replaced by {{tmath|1= a \wedge a = 0 }} for {{tmath|1= a \in V }}.

Note that in the last property above, the real number {{tmath|1= g(a,a) }} need not be nonnegative if {{tmath|1= g }} is not positive-definite. An important property of the geometric product is the existence of elements that have a multiplicative inverse. For a vector {{tmath|1= a }}, if <math>a^2 \ne 0 </math> then <math>a^{-1}</math> exists and is equal to {{tmath|1= g(a,a)^{-1}a }}. A nonzero element of the algebra does not necessarily have a multiplicative inverse. For example, if <math>u</math> is a vector in <math>V</math> such that {{tmath|1= u^2 = 1 }}, the element <math>\textstyle\frac{1}{2}(1 + u)</math> is both a nontrivial [[idempotent element (ring theory)|idempotent element]] and a nonzero [[zero divisor]], and thus has no inverse.{{efn|Given {{tmath|1= u^2 = 1 }}, we have that <math display="inline">(\tfrac{1}{2}(1 + u))^2</math> <math>= \tfrac{1}{4}(1 + 2u + uu)</math> <math> = \tfrac{1}{4}(1 + 2u + 1)</math> {{tmath|1= = \tfrac{1}{2}(1 + u) }}, showing that <math display="inline"> \tfrac{1}{2}(1 + u)</math> is idempotent, and that <math>\tfrac{1}{2}(1 + u)(1 - u)</math> <math>= \tfrac{1}{2}(1 - uu)</math> {{tmath|1= = \tfrac{1}{2}(1 - 1) = 0 }}, showing that it is a nonzero zero divisor.}}

It is usual to identify <math>\R</math> and <math>V</math> with their images under the natural [[embedding]]s <math>\R \to \mathcal{G}(p,q)</math> and {{tmath|1= V \to \mathcal{G}(p,q) }}. In this article, this identification is assumed. Throughout, the terms ''scalar'' and ''vector'' refer to elements of <math>\R</math> and <math>V</math> respectively (and of their images under this embedding).

=== Geometric product ===
{{see also|Symmetric bilinear form|Exterior algebra}}
[[File:GA parallel and perpendicular vectors.svg|200px|right|thumb|Given two vectors <math>a</math> and {{tmath|1= b }}, if the geometric product <math>ab</math> is{{sfn|ps=|Hestenes|2005}} anticommutative; they are perpendicular (top) because {{tmath|1= a \cdot b = 0 }}, if it is commutative; they are parallel (bottom) because {{tmath|1= a \wedge b = 0 }}.]]
{{multiple image
 | left
 | footer = Geometric interpretation of grade-<math>n</math> elements in a real exterior algebra for <math>n = 0</math> (signed point), <math>1</math> (directed line segment, or vector), <math>2</math> (oriented plane element), <math>3</math> (oriented volume). The exterior product of <math>n</math> vectors can be visualized as any {{tmath|1= n }}-dimensional shape (e.g. {{tmath|1= n }}-[[Parallelepiped#Parallelotope|parallelotope]], {{tmath|1= n }}-[[ellipsoid]]); with magnitude ([[hypervolume]]), and [[Orientation (vector space)|orientation]] defined by that on its {{tmath|1= (n - 1) }}-dimensional boundary and on which side the interior is.{{sfn|ps=|Penrose|2007}}{{sfn|ps=|Wheeler|Misner|Thorne|1973|p=83}}
 | width1 = 220
 | image1 = N vector positive.svg
 | caption1 = Orientation defined by an ordered set of vectors.
 | width2 = 220
 | image2 = N vector negative.svg
 | caption2 = Reversed orientation corresponds to negating the exterior product.
}}
For vectors {{tmath|1= a }} and {{tmath|1= b }}, we may write the geometric product of any two vectors {{tmath|1= a }} and {{tmath|1= b }} as the sum of a symmetric product and an antisymmetric product:
: <math>ab = \frac{1}{2} (ab + ba) + \frac{1}{2} (ab - ba) .</math>

Thus we can define the ''inner product'' of vectors as
: <math>a \cdot b := g(a,b),</math>
so that the symmetric product can be written as
: <math>\frac{1}{2}(ab + ba) = \frac{1}{2} \left((a + b)^2 - a^2 - b^2\right) = a \cdot b .</math>

Conversely, {{tmath|1= g }} is completely determined by the algebra. The antisymmetric part is the exterior product of the two vectors, the product of the contained [[exterior algebra]]:
: <math>a \wedge b := \frac{1}{2}(ab - ba) = -(b \wedge a) .</math>

Then by simple addition:
: <math>ab=a \cdot b + a \wedge b </math> the ungeneralized or vector form of the geometric product.

The inner and exterior products are associated with familiar concepts from standard vector algebra. Geometrically, <math>a</math> and <math>b</math> are [[parallel (geometry)|parallel]] if their geometric product is equal to their inner product, whereas <math>a</math> and <math>b</math> are [[perpendicular]] if their geometric product is equal to their exterior product. In a geometric algebra for which the square of any nonzero vector is positive, the inner product of two vectors can be identified with the [[dot product]] of standard vector algebra. The exterior product of two vectors can be identified with the [[signed area]] enclosed by a [[parallelogram]] the sides of which are the vectors. The [[cross product]] of two vectors in <math>3</math> dimensions with positive-definite quadratic form is closely related to their exterior product.

Most instances of geometric algebras of interest have a nondegenerate quadratic form. If the quadratic form is fully [[nondegenerate quadratic form|degenerate]], the inner product of any two vectors is always zero, and the geometric algebra is then simply an exterior algebra. Unless otherwise stated, this article will treat only nondegenerate geometric algebras.

The exterior product is naturally extended as an associative bilinear binary operator between any two elements of the algebra, satisfying the identities
: <math>\begin{align}
                          1 \wedge a_i &= a_i \wedge 1 = a_i \\
  a_1 \wedge a_2\wedge\cdots\wedge a_r &= \frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_r} \operatorname{sgn}(\sigma) a_{\sigma(1)}a_{\sigma(2)} \cdots a_{\sigma(r)},
\end{align}</math>

where the sum is over all permutations of the indices, with <math>\operatorname{sgn}(\sigma)</math> the [[parity of a permutation|sign of the permutation]], and <math>a_i</math> are vectors (not general elements of the algebra). Since every element of the algebra can be expressed as the sum of products of this form, this defines the exterior product for every pair of elements of the algebra. It follows from the definition that the exterior product forms an [[alternating algebra]].

The equivalent structure equation for Clifford algebra is{{sfn|ps=|Wilmot|1988a|p=2338}}{{sfn|ps=|Wilmot|1988b|p=2346}}
: <math> a_1 a_2 a_3 \dots a_n = \sum^{[\frac{n}2]}_{i=0} \sum_{\mu\in{}\mathcal{C}}
(-1)^k \operatorname{Pf}(a_{\mu_1}\cdot a_{\mu_2},\dots,a_{\mu_{2i-1}} \cdot a_{\mu_{2i}})
a_{\mu_{2i+1}}\land\dots\land a_{\mu_n}</math>
where <math>\operatorname{Pf}(A)</math> is the [[Pfaffian]] of {{tmath|1= A }} and <math display="inline">\mathcal{C} = \binom{n}{2i}</math> provides [[combination]]s, {{tmath|1= \mu }}, of {{tmath|1= n }} indices divided into {{tmath|1= 2i }} and {{tmath|1= n - 2i }} parts and {{tmath|1= k }} is the [[parity (mathematics)|parity]] of the [[combination]].

The Pfaffian provides a metric for the exterior algebra and, as pointed out by Claude Chevalley, Clifford algebra reduces to the exterior algebra with a zero quadratic form.{{sfn|ps=|Chevalley|1991}} The role the Pfaffian plays can be understood from a geometric viewpoint by developing Clifford algebra from [[simplex|simplices]].{{sfn|ps=|Wilmot|2023}} This derivation provides a better connection between [[Pascal's triangle]] and [[simplex|simplices]] because it provides an interpretation of the first column of ones.

=== Blades, grades, and basis ===
A multivector that is the exterior product of <math>r</math> linearly independent vectors is called a ''blade'', and is said to be of grade {{tmath|1= r }}.{{efn|Grade is a synonym for ''degree'' of a homogeneous element under the [[graded algebra|grading as an algebra]] with the exterior product (a {{tmath|1= \mathrm{Z} }}-grading), and not under the geometric product.}} A multivector that is the sum of blades of grade <math>r</math> is called a (homogeneous) multivector of grade {{tmath|1= r }}.  From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.

Consider a set of <math>r</math> linearly independent vectors <math>\{a_1,\ldots,a_r\}</math> spanning an {{tmath|1= r }}-dimensional subspace of the vector space. With these, we can define a real [[symmetric matrix]] (in the same way as a [[Gramian matrix]])
: <math>[\mathbf{A}]_{ij} = a_i \cdot a_j</math>

By the [[spectral theorem]], <math>\mathbf{A}</math> can be diagonalized to [[diagonal matrix]] <math>\mathbf{D}</math> by an [[orthogonal matrix]] <math>\mathbf{O}</math> via
: <math>\sum_{k,l}[\mathbf{O}]_{ik}[\mathbf{A}]_{kl}[\mathbf{O}^{\mathrm{T}}]_{lj}=\sum_{k,l}[\mathbf{O}]_{ik}[\mathbf{O}]_{jl}[\mathbf{A}]_{kl}=[\mathbf{D}]_{ij}</math>

Define a new set of vectors {{tmath|1= \{e_1, \ldots,e_r\} }}, known as orthogonal basis vectors, to be those transformed by the orthogonal matrix:
: <math>e_i=\sum_j[\mathbf{O}]_{ij}a_j</math>

Since orthogonal transformations preserve inner products, it follows that <math>e_i\cdot e_j=[\mathbf{D}]_{ij}</math> and thus the <math>\{e_1, \ldots, e_r\}</math> are perpendicular. In other words, the geometric product of two distinct vectors <math>e_i \ne e_j</math> is completely specified by their exterior product, or more generally
: <math>\begin{array}{rl}
e_1e_2\cdots e_r &= e_1 \wedge e_2 \wedge \cdots \wedge e_r \\
&= \left(\sum_j [\mathbf{O}]_{1j}a_j\right) \wedge \left(\sum_j [\mathbf{O}]_{2j}a_j \right) \wedge \cdots \wedge \left(\sum_j [\mathbf{O}]_{rj}a_j\right) \\
&= (\det \mathbf{O}) a_1 \wedge a_2 \wedge \cdots \wedge a_r 
\end{array}</math>

Therefore, every blade of grade <math>r</math> can be written as the exterior product of <math>r</math> vectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by a [[block matrix]] that is orthogonal in the nondegenerate block, and the diagonal matrix has zero-valued entries along the degenerate dimensions. If the new vectors of the nondegenerate subspace are [[unit vector|normalized]] according to
: <math>\widehat{e_i}=\frac{1}{\sqrt{|e_i \cdot e_i|}}e_i,</math>
then these normalized vectors must square to <math>+1</math> or {{tmath|1= -1 }}. By [[Sylvester's law of inertia]], the total number of {{tmath|1= +1 }} and the total number of {{tmath|1= -1 }}s along the diagonal matrix is invariant. By extension, the total number <math>p</math> of these vectors that square to <math>+1</math> and the total number <math>q</math> that square to <math>-1</math> is invariant. (The total number of basis vectors that square to zero is also invariant, and may be nonzero if the degenerate case is allowed.) We denote this algebra {{tmath|1= \mathcal{G}(p,q) }}. For example, <math>\mathcal{G}(3,0)</math> models three-dimensional [[Euclidean space]], <math>\mathcal{G}(1,3)</math> relativistic [[spacetime]] and <math>\mathcal{G}(4,1)</math> a [[conformal geometric algebra]] of a three-dimensional space.

The set of all possible products of <math>n</math> orthogonal basis vectors with indices in increasing order, including <math>1</math> as the empty product, forms a basis for the entire geometric algebra (an analogue of the [[Poincaré–Birkhoff–Witt theorem|PBW theorem]]). For example, the following is a basis for the geometric algebra {{tmath|1= \mathcal{G}(3,0) }}:
: <math>\{1, e_1, e_2, e_3, e_1e_2, e_2e_3, e_3e_1, e_1e_2e_3\}</math>
A basis formed this way is called a '''standard basis''' for the geometric algebra, and any other orthogonal basis for <math>V</math> will produce another standard basis. Each standard basis consists of <math>2^n</math> elements. Every multivector of the geometric algebra can be expressed as a linear combination of the standard basis elements. If the standard basis elements are <math>\{ B_i \mid i \in S \}</math> with <math>S</math> being an index set, then the geometric product of any two multivectors is
: <math> \left( \sum_i \alpha_i B_i \right) \left( \sum_j \beta_j B_j \right) = \sum_{i,j} \alpha_i\beta_j B_i B_j .</math>

The terminology "<math>k</math>-vector" is often encountered to describe multivectors containing elements of only one grade. In higher dimensional space, some such multivectors are not blades (cannot be factored into the exterior product of <math>k</math> vectors). By way of example, <math> e_1 \wedge e_2 + e_3 \wedge e_4 </math> in <math>\mathcal{G}(4,0)</math> cannot be factored; typically, however, such elements of the algebra do not yield to geometric interpretation as objects, although they may represent geometric quantities such as rotations. Only {{tmath|1= 0 }}-, {{tmath|1= 1 }}-, {{tmath|1= (n-1) }}- and {{tmath|1= n }}-vectors are always blades in {{tmath|1= n }}-space.

=== Versor ===
A {{tmath|1= k }}-versor is a multivector that can be expressed as the geometric product of <math>k</math> invertible vectors.{{efn|"reviving and generalizing somewhat a term from hamilton's quaternion calculus which has fallen into disuse" Hestenes defined a {{tmath|1= k }}-versor as a multivector which can be factored into a product of <math>k</math> vectors.{{sfn|ps=|Hestenes|Sobczyk|1984|p=103}}}}{{sfn|ps=|Dorst|Fontijne|Mann|2007|p=204}} Unit quaternions (originally called versors by Hamilton) may be identified with rotors in 3D space in much the same way as real 2D rotors subsume complex numbers; for the details refer to Dorst.{{sfn|ps=|Dorst|Fontijne|Mann|2007|pp=177–182}}

Some authors use the term "versor product" to refer to the frequently occurring case where an operand is "sandwiched" between operators. The descriptions for rotations and reflections, including their outermorphisms, are examples of such sandwiching. These outermorphisms have a particularly simple algebraic form.{{efn|Only the outermorphisms of linear transformations that respect the bilinear form fit this description; outermorphisms are not in general expressible in terms of the algebraic operations.}} Specifically, a mapping of vectors of the form
: <math> V \to V : a \mapsto RaR^{-1}</math> extends to the outermorphism <math>\mathcal{G}(V) \to \mathcal{G}(V) : A \mapsto RAR^{-1}.</math>

Since both operators and operand are versors there is potential for alternative examples such as rotating a rotor or reflecting a spinor always provided that some geometrical or physical significance can be attached to such operations.

By the [[Cartan–Dieudonné theorem]] we have that every isometry can be given as reflections in hyperplanes and since composed reflections provide rotations then we have that orthogonal transformations are versors.

In group terms, for a real, non-degenerate {{tmath|1= \mathcal{G}(p,q) }}, having identified the group <math>\mathcal{G}^\times</math> as the group of all invertible elements of {{tmath|1= \mathcal{G} }}, Lundholm gives a proof that the "versor group" <math>\{ v_1 v_2 \cdots v_k \in \mathcal{G} \mid v_i \in V^\times\}</math> (the set of invertible versors) is equal to the Lipschitz group <math>\Gamma</math> ({{aka}} Clifford group, although Lundholm deprecates this usage).{{sfn|ps=|Lundholm|Svensson|2009|pp=58 ''et seq''}}

=== Subgroups of the Lipschitz group ===
We denote the grade involution as {{tmath|1= \widehat{S} }} and reversion as {{tmath|1= \widetilde{S} }}.

Although the Lipschitz group (defined as {{tmath|1= \{ S \in \mathcal{G}^{\times} \mid \widehat{S} V S^{-1} \subseteq V \} }}) and the versor group (defined as {{tmath|1= \textstyle \{ \prod_{i=0}^{k} v_i \mid v_i \in V^{\times}, k \in \N \} }}) have divergent definitions, they are the same group.  Lundholm defines the {{tmath|1= \operatorname{Pin} }}, {{tmath|1= \operatorname{Spin} }}, and {{tmath|1= \operatorname{Spin}^{+} }} subgroups of the Lipschitz group.{{sfn|ps=|Lundholm|Svensson|2009|p=58}}

{| class="wikitable"
|-
 ! Subgroup !! Definition !! GA term
|-
 | <math>\Gamma</math> || <math> \{ S \in \mathcal{G}^{\times} \mid \widehat{S} V S^{-1} \subseteq V \} </math> || versors
|-
 | <math>\operatorname{Pin}</math> || <math> \{ S \in \Gamma \mid S \widetilde{S} = \pm 1 \} </math> || unit versors
|-
 | <math>\operatorname{Spin}</math> || <math> {\operatorname{Pin}} \cap \mathcal{G}^{[0]} </math> || even unit versors
|-
 | <math>\operatorname{Spin}^{+}</math> || <math> \{ S \in \operatorname{Spin} \mid S \widetilde{S} = 1 \} </math> || rotors
|-
|}

Multiple analyses of spinors use GA as a representation.{{sfn|ps=|Francis|Kosowsky|2008}}

=== 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) }}.

=== Representation of subspaces ===
{{see also|Grassmannian}}
Geometric algebra represents subspaces of <math>V</math> as blades, and so they coexist in the same algebra with vectors from {{tmath|1= V }}. A {{tmath|1= k }}-dimensional subspace <math>W</math> of <math>V</math> is represented by taking an orthogonal basis <math>\{b_1,b_2,\ldots, b_k\}</math> and using the geometric product to form the [[blade (geometry)|blade]] {{tmath|1= D = b_1b_2\cdots b_k }}. There are multiple blades representing {{tmath|1= W }}; all those representing <math>W</math> are scalar multiples of {{tmath|1= D }}. These blades can be separated into two sets: positive multiples of <math>D</math> and negative multiples of {{tmath|1= D }}. The positive multiples of <math>D</math> are said to have ''the same [[orientation (vector space)|orientation]]'' as {{tmath|1= D }}, and the negative multiples the ''opposite orientation''.

Blades are important since geometric operations such as projections, rotations and reflections depend on the factorability via the exterior product that (the restricted class of) {{tmath|1= n }}-blades provide but that (the generalized class of) grade-{{tmath|1= n}} multivectors do not when {{tmath|1= n \ge 4 }}.

=== 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.

=== Extensions of the inner and exterior products ===
It is common practice to extend the exterior product on vectors to the entire algebra. This may be done through the use of the above-mentioned [[#Grade projection|grade projection]] operator:
: <math>C \wedge D := \sum_{r,s}\langle \langle C \rangle_r \langle D \rangle_s \rangle_{r+s} </math> &nbsp; &nbsp; (the ''exterior product'')

This generalization is consistent with the above definition involving antisymmetrization. Another generalization related to the exterior product is the commutator product:
: <math>C \times D := \tfrac{1}{2}(CD-DC) </math> &nbsp; &nbsp; (the ''commutator product'')

The regressive product is the dual of the exterior product (respectively corresponding to the "meet" and "join" in this context).{{efn|[...] the exterior product operation and the join relation have essentially the same meaning. The [[Grassmann–Cayley algebra]] regards the meet relation as its counterpart and gives a unifying framework in which these two operations have equal footing [...] Grassmann himself defined the meet operation as the dual of the exterior product operation, but later mathematicians defined the meet operator independently of the exterior product through a process called [[Shuffle algebra|shuffle]], and the meet operation is termed the shuffle product. It is shown that this is an antisymmetric operation that satisfies associativity, defining an algebra in its own right. Thus, the Grassmann–Cayley algebra has two algebraic structures simultaneously: one based on the exterior product (or join), the other based on the shuffle product (or meet). Hence, the name "double algebra", and the two are shown to be dual to each other.{{sfn|ps=|Kanatani|2015|pp=112–113}}}} The dual specification of elements permits, for blades {{tmath|1= C }} and {{tmath|1= D }}, the intersection (or meet) where the duality is to be taken relative to the a blade containing both {{tmath|1= C }} and {{tmath|1= D }} (the smallest such blade being the join).{{sfn|ps=|Dorst|Lasenby|2011|p=443}}
: <math>C \vee D := ((CI^{-1}) \wedge (DI^{-1}))I </math>
with {{tmath|1= I }} the unit pseudoscalar of the algebra. The regressive product, like the exterior product, is associative.{{sfn|ps=|Vaz|da Rocha|2016|loc=§2.8}}

The inner product on vectors can also be generalized, but in more than one non-equivalent way. The paper {{Harvard citation|Dorst|2002}} gives a full treatment of several different inner products developed for geometric algebras and their interrelationships, and the notation is taken from there. Many authors use the same symbol as for the inner product of vectors for their chosen extension (e.g. Hestenes and Perwass). No consistent notation has emerged.

Among these several different generalizations of the inner product on vectors are:
: <math> C \;\rfloor\; D := \sum_{r,s}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{s-r} </math> &nbsp;&nbsp;(the ''left contraction'')
: <math> C \;\lfloor\; D := \sum_{r,s}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{r-s} </math> &nbsp;&nbsp;(the ''right contraction'')
: <math> C * D := \sum_{r,s}\langle \langle C \rangle_r \langle D \rangle_s \rangle_{0} </math> &nbsp;&nbsp;(the ''scalar product'')
: <math> C \bullet D := \sum_{r,s}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{|s-r|} </math> &nbsp;&nbsp;(the "(fat) dot" product){{efn|
This should not be confused with Hestenes's irregular generalization {{tmath|1= \textstyle C \bullet_\text{H} D := \sum_{r\ne0,s\ne0}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{ \vert s-r \vert } }}, where the distinguishing notation is from {{harvp|Dorst|Fontijne|Mann|2007|p=590|loc=§B.1, which makes the point that scalar components must be handled separately with this product.}}}}

{{harvtxt|Dorst|2002}} makes an argument for the use of contractions in preference to Hestenes's inner product; they are algebraically more regular and have cleaner geometric interpretations. 
A number of identities incorporating the contractions are valid without restriction of their inputs.
For example,
: <math> C \;\rfloor\; D = ( C \wedge ( D I^{-1} ) ) I </math>
: <math> C \;\lfloor\; D = I ( ( I^{-1} C) \wedge D ) </math>
: <math> ( A \wedge B ) * C = A * ( B \;\rfloor\; C ) </math>
: <math> C * ( B \wedge A ) = ( C \;\lfloor\; B ) * A </math> 
: <math> A \;\rfloor\; ( B \;\rfloor\; C ) = ( A \wedge B ) \;\rfloor\; C </math>
: <math> ( A \;\rfloor\; B ) \;\lfloor\; C = A \;\rfloor\; ( B \;\lfloor\; C ) .</math>

Benefits of using the left contraction as an extension of the inner product on vectors include that the identity <math> ab = a \cdot b + a \wedge b </math> is extended to <math> aB = a \;\rfloor\; B + a \wedge B</math> for any vector <math>a</math> and multivector {{tmath|1= B }}, and that the [[projection (linear algebra)|projection]] operation <math> \mathcal{P}_b (a) = (a \cdot b^{-1})b </math> is extended to <math> \mathcal{P}_B (A) = (A \;\rfloor\; B^{-1}) \;\rfloor\; B</math> for any blade <math>B</math> and any multivector <math>A</math> (with a minor modification to accommodate null {{tmath|1= B }}, given [[#Projection and rejection|below]]).

=== Dual basis ===
Let <math>\{ e_1 , \ldots , e_n \}</math> be a basis of {{tmath|1= V }}, i.e. a set of <math>n</math> linearly independent vectors that span the {{tmath|1= n }}-dimensional vector space {{tmath|1= V }}. The basis that is dual to <math>\{ e_1 , \ldots , e_n \}</math> is the set of elements of the [[dual vector space]] <math>V^{*}</math> that forms a [[biorthogonal system]] with this basis, thus being the elements denoted <math>\{ e^1 , \ldots , e^n \}</math> satisfying
: <math>e^i \cdot e_j = \delta^i{}_j,</math>
where <math>\delta</math> is the [[Kronecker delta]].

Given a nondegenerate quadratic form on {{tmath|1= V }}, <math>V^{*}</math> becomes naturally identified with {{tmath|1= V }}, and the dual basis may be regarded as elements of {{tmath|1= V }}, but are not in general the same set as the original basis.

Given further a GA of {{tmath|1= V }}, let 
: <math>I = e_1 \wedge \cdots \wedge e_n</math>
be the pseudoscalar (which does not necessarily square to {{tmath|1= \pm 1 }}) formed from the basis {{tmath|1= \{ e_1 , \ldots , e_n \} }}. The dual basis vectors may be constructed as
: <math>e^i=(-1)^{i-1}(e_1 \wedge \cdots \wedge \check{e}_i \wedge \cdots \wedge e_n) I^{-1},</math>
where the <math>\check{e}_i</math> denotes that the {{tmath|1= i }}th basis vector is omitted from the product.

A dual basis is also known as a [[reciprocal basis]] or reciprocal frame.

A major usage of a dual basis is to separate vectors into components. Given a vector {{tmath|1= a }}, scalar components <math>a^i</math> can be defined as
: <math>a^i=a\cdot e^i\ ,</math>
in terms of which <math>a</math> can be separated into vector components as
: <math>a=\sum_i a^i e_i\ .</math>
We can also define scalar components <math>a_i</math> as
: <math>a_i=a\cdot e_i\ ,</math>
in terms of which <math>a</math> can be separated into vector components in terms of the dual basis as
: <math>a=\sum_i a_i e^i\ .</math>

A dual basis as defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra.{{sfn|ps=|Hestenes|Sobczyk|1984|p=31}} For compactness, we'll use a single capital letter to represent an ordered set of vector indices. I.e., writing
: <math>J=(j_1,\dots ,j_n)\ ,</math>
where {{tmath|1= j_1 < j_2 < \dots < j_n }},
we can write a basis blade as
: <math>e_J=e_{j_1}\wedge e_{j_2}\wedge\cdots\wedge e_{j_n}\ .</math>
The corresponding reciprocal blade has the indices in opposite order:
: <math>e^J=e^{j_n}\wedge\cdots \wedge e^{j_2}\wedge e^{j_1}\ .</math>
Similar to the case above with vectors, it can be shown that
: <math>e^J * e_K=\delta^J_K\ ,</math>
where <math>*</math> is the scalar product.

With <math>A</math> a multivector, we can define scalar components as{{sfn|ps=|Doran|Lasenby|2003|p=102}}
: <math>A^{ij\cdots k}=(e^k\wedge\cdots\wedge e^j\wedge e^i)*A\ ,</math>
in terms of which <math>A</math> can be separated into component blades as
: <math>A=\sum_{i<j<\cdots<k} A^{ij\cdots k} e_i\wedge e_j\wedge\cdots \wedge e_k\ .</math>
We can alternatively define scalar components
: <math>A_{ij\cdots k}=(e_k\wedge\cdots\wedge e_j\wedge e_i)*A\ ,</math>
in terms of which <math>A</math> can be separated into component blades as
: <math>A=\sum_{i<j<\cdots<k} A_{ij\cdots k} e^i\wedge e^j\wedge\cdots \wedge e^k\ .</math>

=== Linear functions ===
Although a versor is easier to work with because it can be directly represented in the algebra as a multivector, versors are a subgroup of [[linear function]]s on multivectors, which can still be used when necessary. The geometric algebra of an {{tmath|1= n }}-dimensional vector space is spanned by a basis of <math>2^n</math> elements. If a multivector is represented by a <math>2^n \times 1</math> real [[column matrix]] of coefficients of a basis of the algebra, then all linear transformations of the multivector can be expressed as the [[matrix multiplication]] by a <math>2^n \times 2^n</math> real matrix. However, such a general linear transformation allows arbitrary exchanges among grades, such as a "rotation" of a scalar into a vector, which has no evident geometric interpretation.

A general linear transformation from vectors to vectors is of interest. With the natural restriction to preserving the induced exterior algebra, the ''[[outermorphism]]'' of the linear transformation is the unique{{efn|The condition that <math>\underline{\mathsf{f}}(1) = 1</math> is usually added to ensure that the [[zero map]] is unique.}} extension of the versor. If <math>f</math> is a linear function that maps vectors to vectors, then its outermorphism is the function that obeys the rule
: <math>\underline{\mathsf{f}}(a_1 \wedge a_2 \wedge \cdots \wedge a_r) = f(a_1) \wedge f(a_2) \wedge \cdots \wedge f(a_r)</math>
for a blade, extended to the whole algebra through linearity.