Editing Geometric algebra (section)

=== 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.