Affine Coordinates are Well-Defined

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Theorem

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

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:

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

Then $\Theta_\mathcal R$ is a bijection.


Proof

Proof of Surjection

Let $p \in \mathcal E$.

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

Let $\left({\lambda_1, \ldots, \lambda_n}\right)$ be coordinates of $v$ in the basis $\left({e_1, \ldots, e_n}\right)$.

Then:

\(\displaystyle p_0 + \sum_{i \mathop = 1}^n \lambda_ie_i\) \(=\) \(\displaystyle p_0 + v\)
\(\displaystyle \) \(=\) \(\displaystyle p_0 + \left({p - p_0}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle p\)

thus demonstrating that $\Theta_\mathcal R$ is a surjection.


Proof of Injection

Let:

$\Theta_\mathcal R \left({\lambda_1, \ldots, \lambda_n}\right) = \Theta_\mathcal R \left({\mu_1, \ldots, \mu_n}\right)$

That is:

$\displaystyle p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i = p_0 + \sum_{i \mathop = 1}^n \mu_i e_i$

Then by $(3)$ of Properties of Affine Spaces:

$\displaystyle \sum_{i \mathop = 1}^n \lambda_i e_i = \sum_{i \mathop = 1}^n \mu_i e_i$

By Expression of Vector as Linear Combination from Basis is Unique:

$\lambda_i = \mu_i$

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

Hence the result.

$\blacksquare$