Affine space/Affine Basis/Barycentric coordinates/Fact/Proof

Proof

Let ${}i_{0}\in I$ be fixed. In ${}V$, we have a unique representation

${}{\overrightarrow {P_{i_{0}}P}}=\sum _{i\in I\setminus \{i_{0}\}}a_{i}{\overrightarrow {P_{i_{0}}P_{i}}}\,.$

We set

${}a_{i_{0}}:=1-\sum _{i\in I\setminus \{i_{0}\}}a_{i}\,.$

Then ${}\sum _{i\in I}a_{i}=1$, and

${}{\begin{aligned}P&=P_{i_{0}}+{\overrightarrow {P_{i_{0}}P}}\\&=P_{i_{0}}+\sum _{i\in I\setminus \{i_{0}\}}a_{i}{\overrightarrow {P_{i_{0}}P_{i}}}\\&=P_{i_{0}}+a_{i_{0}}{\overrightarrow {P_{i_{0}}P_{i_{0}}}}+\sum _{i\in I\setminus \{i_{0}\}}a_{i}{\overrightarrow {P_{i_{0}}P_{i}}}\\&=P_{i_{0}}+\sum _{i\in I}a_{i}{\overrightarrow {P_{i_{0}}P_{i}}}.\end{aligned}}$

Therefore, there exists such a representation with ${}P_{i_{0}}$ as origin. Uniqueness follows from the facts that the ${}a_{i}$, ${}i\neq i_{0}$, are uniquely determined as the coefficients of the vector space basis, and that ${}a_{i_{0}}$ is determined by the baryzentric condition.

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