Today I Learned

Some of the things I've learned every day since Oct 10, 2016

74: Linearly Separable Values (Euclidean)

2 sets X_1, X_2 of points in n-dimensional Euclidean space E^n are linearly separable if and only if there exists a non-zero vector \mathbf{w} \in E^n and a number k such that

\mathbf{w} \cdot \mathbf{x} < k

holds for every \mathbf{x} \in X_1, and does not hold for every \mathbf{x'} \in X_2. Intuitively, this means that two sets of points in an n-dimensional Euclidean space are linearly separable if there is an (n-1)-dimensional plane that when inserted into the same space separates the two sets.

(This concept could probably be extended to spaces which share certain properties with E^n, such as having a partial order, closure, etc., but E^n gives the simplest example.)

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