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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $P$ be an [[affine space|09XNAPTJQ7]] over $V$.\nLet $p_0, \\dots, p_k$ be points of $P$.\nThey are *affinely independent* if the vectors $p_1 - p_0, \\dots, p_k - p_0$ of $V$ are [[linearly independent|0CRMGPAH83]].\n\n## Remarks\n\nExample: in $\\R^2$, the points $(0,0)$, $(1,0)$, $(0,1)$ are affinely independent, since the vectors $(1,0)$ and $(0,1)$ are linearly independent.\nNon-example: the points $(0,0)$, $(1,1)$, $(2,2)$ are not, since $(2,2) = 2 \\cdot (1,1)$.\nThe condition does not depend on which point plays the role of $p_0$.\nAn equivalent form: for all $\\lambda_0, \\dots, \\lambda_k \\in F$ with $\\sum_i \\lambda_i = 0$ and $\\sum_i \\lambda_i (p_i - p_0) = 0$, all $\\lambda_i$ are $0$.\nAnother equivalent form: no $p_j$ is an [[affine combination|0KJH6SRFZK]] of the other points.\nIn an affine space of dimension $n$, at most $n + 1$ points can be affinely independent.\n",
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  "name": "Affine independence",
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