Editing Geometric algebra (section)

=== Vector space model ===
{{main | Comparison of vector algebra and geometric algebra}}

The [[even subalgebra]] of <math>\mathcal{G}(2,0)</math> is isomorphic to the [[complex number]]s, as may be seen by writing a vector <math>P</math> in terms of its components in an orthonormal basis and left multiplying by the basis vector {{tmath|1= e_1 }}, yielding
: <math> Z = e_1 P = e_1 ( x e_1 + y e_2) = x (1) + y ( e_1 e_2) ,</math>
where we identify <math>i \mapsto e_1e_2</math> since
: <math>(e_1 e_2)^2 = e_1 e_2 e_1 e_2 = -e_1 e_1 e_2 e_2 = -1 .</math>

Similarly, the even subalgebra of <math>\mathcal{G}(3,0)</math> with basis <math>\{1, e_2 e_3, e_3 e_1, e_1 e_2 \}</math> is isomorphic to the [[quaternion]]s as may be seen by identifying {{tmath|1= i \mapsto -e_2 e_3 }}, <math>j \mapsto -e_3 e_1</math> and {{tmath|1= k \mapsto -e_1 e_2 }}.

Every [[associative algebra]] has a matrix representation; replacing the three Cartesian basis vectors by the [[Pauli matrices]] gives a representation of {{tmath|1= \mathcal{G}(3,0) }}:
: <math>\begin{align}
 e_1 = \sigma_1 = \sigma_x &=
 \begin{pmatrix}
  0 & 1 \\
  1 & 0
 \end{pmatrix} \\
 e_2 = \sigma_2 = \sigma_y &=
 \begin{pmatrix}
  0 & -i \\
  i & 0
 \end{pmatrix} \\
 e_3 =\sigma_3 = \sigma_z &=
 \begin{pmatrix}
  1 & 0 \\
  0 & -1
 \end{pmatrix} \,.
\end{align}</math>

Dotting the "[[Pauli matrices#Pauli vectors|Pauli vector]]" (a [[dyadics|dyad]]):
: <math>\sigma = \sigma_1 e_1 + \sigma_2 e_2 + \sigma_3 e_3</math> with arbitrary vectors <math> a </math> and <math> b </math> and multiplying through gives:
: <math>(\sigma \cdot a)(\sigma \cdot b) = a \cdot b + a \wedge b </math> (Equivalently, by inspection, {{tmath|1= a \cdot b + i \sigma \cdot ( a \times b ) }})