Editing Linear separability (section)

==Mathematical definition==

Let <math>X_{0}</math> and <math>X_{1}</math> be two sets of points in an ''n''-dimensional Euclidean space. Then <math>X_{0}</math> and <math>X_{1}</math> are ''linearly separable'' if there exist ''n'' + 1 real numbers <math>w_{1}, w_{2},..,w_{n}, k</math>, such that every point <math>x \in X_{0}</math> satisfies <math>\sum^{n}_{i=1} w_{i}x_{i} > k</math> and every point <math>x \in X_{1}</math> satisfies <math>\sum^{n}_{i=1} w_{i}x_{i} < k</math>, where <math>x_{i}</math> is the <math>i</math>-th component of <math>x</math>.

Equivalently, two sets are linearly separable precisely when their respective [[convex hull]]s are [[disjoint sets|disjoint]] (colloquially, do not overlap).<ref>{{Cite book |last1=Boyd |first1=Stephen |url=http://dx.doi.org/10.1017/cbo9780511804441 |title=Convex Optimization |last2=Vandenberghe |first2=Lieven |date=2004-03-08 |publisher=Cambridge University Press |doi=10.1017/cbo9780511804441 |isbn=978-0-521-83378-3}}</ref>

In simple 2D, it can also be imagined that the set of points under a linear transformation collapses into a line, on which there exists a value, k, greater than which one set of points will fall into, and lesser than which the other set of points fall.