Definition:Vectorialization of Affine Space

Definition

Let $\EE$ be an affine space over a field $k$ with difference space $E$.

Let $\RR = \tuple {p_0, e_1, \ldots, e_n}$ be an affine frame in $\EE$.

Define a mapping $\Theta_\RR : k^n \to \EE$ by:

$\ds \map {\Theta_\RR} {\lambda_1, \ldots, \lambda_n} = p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i$.

By Affine Coordinates are Well-Defined, $\Theta_\RR$ is a bijection.

For any $\mu \in k$, $p, q \in \EE$ let:

$\mu \cdot p = \map {\Theta_\RR} {\mu \cdot \map {\Theta_\RR^{-1} } p}$

and:

$p + q = \map {\Theta_\RR} {\map {\Theta_\RR^{-1} } p + \map {\Theta_\RR^{-1} } q}$

We call the set $\EE$, together with the operations $\cdot, +$ the vectorialization of $\EE$ with origin $p_0$.

Also see

• Vectorialization of Affine Space is Vector Space