Affine Coordinates are Well-Defined

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

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

Define a mapping $\Theta_\RR : k^n \to \EE$ by:
 * $\ds \map {\Theta_\RR} {\lambda_1, \ldots, \lambda_n} = p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i$

Then $\Theta_\RR$ is a bijection.

Proof of Surjection
Let $p \in \EE$.

Let $v = p - p_0 \in V$.

Let $\tuple {\lambda_1, \ldots, \lambda_n}$ be coordinates of $v$ in the basis $\tuple {e_1, \ldots, e_n}$.

Then:

thus demonstrating that $\Theta_\RR$ is a surjection.

Proof of Injection
Let:
 * $\map {\Theta_\RR} {\lambda_1, \ldots, \lambda_n} = \map {\Theta_\RR} {\mu_1, \ldots, \mu_n}$

That is:
 * $\ds p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i = p_0 + \sum_{i \mathop = 1}^n \mu_i e_i$

Then by $(3)$ of Properties of Affine Spaces:
 * $\ds \sum_{i \mathop = 1}^n \lambda_i e_i = \sum_{i \mathop = 1}^n \mu_i e_i$

By Expression of Vector as Linear Combination from Basis is Unique:
 * $\lambda_i = \mu_i$

for $i = 1, \ldots, n$.

Hence the result.