Affine space/Affine generating system/Introduction/Section

Lemma

Let ${}E$ be an affine space over the ${}K$-vector space ${}V$. In this situation, the intersection of a family of affine subspaces ${}F_{i}\subseteq E$, ${}i\in I$,

is again an affine subspace.

Proof

If the intersection is empty, then the statement holds by definition. So let ${}P\in \bigcap _{i\in I}F_{i}$. We may write the affine subspaces as

${}F_{i}=P+U_{i}\,$

with linear subspaces ${}U_{i}\subseteq V$. Let

${}U=\bigcap _{i\in I}U_{i}\,,$

which is a linear subspace, due to fact  (1). We claim that

${}\bigcap _{i\in I}F_{i}=P+U\,.$

From ${}Q\in \bigcap _{i\in I}F_{i}$, we can deduce

${}Q=P+u\,$

with ${}u\in \bigcap _{i\in I}U_{i}$, so that ${}Q\in P+U$ holds. If ${}Q\in P+U$ holds, then directly ${}Q\in P+U_{i}=F_{i}$ follows.

$\Box$

In particular, for every subset ${}T\subseteq E$ in an affine space ${}E$, there exist a smallest affine subspace containing ${}T$.

Lemma

Let ${}E$ be an affine space over the ${}K$-vector space ${}V$, and let ${}T\subseteq E$ denote a subset. Then, the smallest affine subspace ${}F\subseteq E$ of ${}E$, which contains ${}T$, consists of all barycentric combinations

$\sum _{i=1}^{n}a_{i}P_{i}{\text{ with }}P_{i}\in T{\text{ and }}\sum _{i=1}^{n}a_{i}=1.$

Proof

The given set contains the points from ${}T$, as we can take a standard tuple as a barycentric coordinate tuple. Therefore, the claim follows from fact and exercise.

$\Box$

Definition

Let ${}E$ be an affine space over the ${}K$-vector space ${}V$, and let ${}F\subseteq E$ be an affine subspace. A family of points ${}P_{i}\in F$, ${}i\in I$, is called an affine generating system

of ${}F$, if ${}F$ is the smallest affine subspace of ${}E$ containing all points ${}P_{i}$.

A point generates, as an affine space, the point itself, two points generate the connecting line.