Editing Geometric algebra (section)

=== 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>