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b 0 + b 1 x 1 + ... + b p x p = 0 (19.1)
b 0 ≠ 0 gives us an affine plane, which does not pass through the origin. We can use a more
succinct notation for this equation by introducing the summation sign:
p∑
b 0 + b j x j = 0 (19.2)
j=1
Notice however that this is nothing more than a multi-dimensional dot product (or, more
generally, an inner product) and as such can be written even more succinctly as:
b · x + b 0 = 0 (19.3)
If an element x ∈ R p satisfies this relation then it lives on the p − 1-dimensional hyperplane.
This hyperplane splits the p-dimensional feature space into two classification regions. This can be
seen in Figure 19.2. Compare this to the hyperblock partitioning of the Decision Tree displayed
in the previous chapter:
Figure 19.2: Separation of p-dimensional space by a hyperplane
Elements x above the plane satisfy:
b · x + b 0 > 0 (19.4)
While those below it satisfy:
b · x + b 0 < 0 (19.5)