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Definition: A set (x_0, ... , x_k} of points of R^N is said to be geometrically independent, or affinely independent, if the equations sum[(a_i)(x_i)]=0 (with i=0 to i=k), and sum(a_i)=0 (with i=0 to i=k) hold only if each a_i = 0.
Source: Topology (second edition) by James R. Munkres