 (In other words, formula_4 is a one-form or linear functional mapping formula_1 onto R.) The weight vector formula_3 is learned from a set of labeled training samples.Often "f" is a threshold function, which maps all values of formula_7 above a certain threshold to the first class and all other values to the second class; e.g., 

A more complex "f" might give the probability that an item belongs to a certain class.For a two-class classification problem, one can visualize the operation of a linear classifier as splitting a high-dimensional input space with a hyperplane