# Definition:Vectorialization of Affine Space

## Definition

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

Let $\mathcal R = \left({p_0, e_1, \ldots, e_n}\right)$ be an affine frame in $\mathcal E$.

Define a mapping $\Theta_{\mathcal R} : k^n \to \mathcal E$ by:

$\displaystyle \Theta_{\mathcal R} \left({\lambda_1, \ldots, \lambda_n}\right) = p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i$.

By Affine Coordinates are Well-Defined, $\Theta_{\mathcal R}$ is a bijection.

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

$\mu \cdot p = \Theta_{\mathcal R}\left({ \mu \cdot \Theta_{\mathcal R}^{-1} \left({p}\right) }\right)$

and:

$p + q = \Theta_{\mathcal R} \left({ \Theta_{\mathcal R}^{-1} \left({p}\right) + \Theta_{\mathcal R}^{-1} \left({q}\right) }\right)$

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