# Affine space/Affine bases/Introduction/Section
## Definition

A family of points $P_{i}\in E$, $i\in I$, in an affine space $E$ over a $K$-vector space $V$ is called an affine basis of $E$, if there exists an $i_{0}\in I$ such that the family of vectors
    ${\overrightarrow {P_{i_{0}}P_{i}}},i\in I\setminus \{i_{0}\},$
is a basis
$V$
Because of
    ${\overrightarrow {P_{i_{0}}P_{i}}}={\overrightarrow {P_{i_{0}}P_{i_{1}}}}+{\overrightarrow {P_{i_{1}}P_{i}}}=-{\overrightarrow {P_{i_{1}}P_{i_{0}}}}+{\overrightarrow {P_{i_{1}}P_{i}}}\,,$
the basis vectors ${\overrightarrow {P_{i_{0}}P_{i}}}$ with respect to the origin $P_{i_{0}}$ can be expressed as linear combinations of the corresponding vectors with respect to any other origin point $P_{i_{1}}$ of the family. Therefore, the property of being an affine basis is independent from the chosen $P_{i_{0}}$.

## Definition

For a family $P_{i}$, $i\in I$, of points in an affine space $E$ and a tuple $a_{i}$, $i\in I$, in $K$ satisfying
    $\sum _{i\in I}a_{i}=1\,$
(for $I$ infinite, only finitely many of the $a_{i}$ are allowed to be different from $0$), the sum $\sum _{i\in I}a_{i}P_{i}$ is called a barycentric combination of the $P_{i}$. The corresponding point in $E$ is given by
    $\sum _{i\in I}a_{i}P_{i}=Q+\sum _{i\in I}a_{i}{\overrightarrow {QP_{i}}}\,,$
$Q$
$E$

## Lemma

For a family $P_{i}$, $i\in I$, of points in an affine space $E$, a
barycentric combination
    $\sum _{i\in I}a_{i}P_{i}$
$E$

### Proof

$\Box$

## Theorem

Let $P_{i}$, $i\in I$, denote an affine basis in an affine space $E$ over the $K$-vector space $V$. Then, for every point $P\in E$, there exists a unique barycentric representation
    $P=\sum _{i\in I}a_{i}P_{i}\,.$

### 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.
$\Box$

## Definition

Let $P_{i}$, $i\in I$, denote an affine basis in an affine space $E$ over the $K$-vector space $V$. For a point $P\in E$, the uniquely determined numbers
    $(a_{i},i\in I){\text{ with }}\sum _{i\in I}a_{i}=1$
such that
    $P=\sum _{i\in I}a_{i}P_{i}\,$
holds, are called the barycentric coordinates
$P$

## Example

Let $P_{i}$, $i\in I$, be an affine basis in an affine space $E$ over the $K$-vector space $V$. Then the point $P_{j}$ ($j=I$) has the barycentric coordinates $(0,\ldots ,0,1,0,\ldots ,0)$, where the $1$ is at the $j$-th place ($I$ being finite and ordered).