Affine space/Affinely independence/Introduction/Section

Definition

Let ${}E$ be an affine space over a ${}K$-vector space ${}V$, and let

$P_{1},\ldots ,P_{n}$

be a finite family of points in ${}E$. We say that this family of points is affinely independent, if an equality

${}\sum _{i=1}^{n}a_{i}P_{i}=\sum _{i=1}^{n}b_{i}P_{i}\,$

wit

${}\sum _{i=1}^{n}a_{i}=\sum _{i=1}^{n}b_{i}=1\,$

is only possible if

${}a_{i}=b_{i}\,$

for all

${}i=1,\ldots ,n$.

Lemma

Let ${}E$ be an affine space over a ${}K$-vector space ${}V$, and let

$P_{1},\ldots ,P_{n}$

denote a finite family of points in ${}E$. Then the following statements are equivalent.

1. The points ${}P_{1},\ldots ,P_{n}$ are affinely independent.
2. For every ${}i\in \{1,\ldots ,n\}$, the family of vectors

   ${\overrightarrow {P_{i}P_{1}}},\ldots ,{\overrightarrow {P_{i}P_{i-1}}},\,{\overrightarrow {P_{i}P_{i+1}}},\ldots ,{\overrightarrow {P_{i}P_{n}}}$

   is linearly independent.

3. There exists some ${}i\in \{1,\ldots ,n\}$ such that the family of vectors

   ${\overrightarrow {P_{i}P_{1}}},\ldots ,{\overrightarrow {P_{i}P_{i-1}}},\,{\overrightarrow {P_{i}P_{i+1}}},\ldots ,{\overrightarrow {P_{i}P_{n}}}$

   is linearly independent.

4. The points ${}P_{1},\ldots ,P_{n}$ form an affine basis in the affine subspace generated by them.

Proof

See exercise.

$\Box$