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General position

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A collection of $n+1$ points $$v_0, v_1, \ldots, v_n$$ in a vector space is said to be in general position if

$v_1 - v_0, \ldots, v_n - v_0$ are linearly independent.

As shown here:

General position in R3.jpg

The definition is independent from the order.

The simplest example of this occurring is an $n$-simplex in ${\bf R}^n$. Its $n+1$ vertices are at $$(0,0,0,0,...,0), (1,0,0,0,...,0), (0,1,0,0,...,0), \ldots , (0,0,0,0,...0,1,0), (0,0,0,0,...,0,1).$$ Illustrated in 3D:

3-simplex.jpg

In general, we use the notation $$v_0 v_1 \ldots v_n = {\rm conv}\{v_0, v_1, \ldots, v_n \}.$$