What does it mean to be "affinely independent", and why is it important to learn?

affine-geometry

Solution 1:

Roughly speaking, affine independence is like linear independence but without the restriction that the subset of lower dimension the points lie in contains the origin. So three points in space are affinely independent if the smallest flat thing containing them is a plane. They're affinely dependent if they lie on a line (or are the same point).

A set of points is affinely dependent if and only if when you subtract one of them from the others the resulting set (excluding the $0$ vector that results from subtracting the one you chose from itself) is linearly dependent.

The language of affine independence is useful if you don't really care where the origin is in your representation of $n$-space. That might be the case if the points are vectors of $n$ numerical attributes, one vector for each participant in a survey. The page you link to suggests "free vectors" in physics as another motivation for affine geometry.

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