Vectorialization of Affine Space is Vector Space

Theorem
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 $\EE$.

Let $\struct {\EE, +, \cdot}$ be the vectorialization of $\EE$.

Then $\struct {\EE, +, \cdot}$ is a vector space.

Proof
By the definition of the vectorialization of an affine space, the mapping $\Theta_\RR : K^n \to \EE$ defined by:
 * $\ds \map {\Theta_\RR} {\lambda_1, \ldots, \lambda_n} = p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i$

is a bijection from $K^n$ to $\EE$.

Therefore, by Homomorphic Image of Vector Space, it suffices to prove that $\Theta_\RR$ is a linear transformation.

By General Linear Group is Group:
 * $\Theta_\RR$ is a linear transformation its inverse ${\Theta_\RR}^{-1}$ is a linear transformation.

Therefore, it suffices to show that:
 * $\forall p, q \in \EE, \mu \in K: \map { {\Theta_\RR}^{-1} } {\mu \cdot p + q} = \mu \cdot \map { {\Theta_\RR}^{-1} } p + \map { {\Theta_\RR}^{-1} } g$

Thus:

This is the required identity.