Affine space/Finite point family/Affinely independent/Characterization/Fact/Proof/Exercise

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 ${}E$. Show that 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.

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