9.1.1 What Is a Hyperplane?

In a p -dimensional space, a hyperplane is a flat affine subspace of hyperplane dimension p − 1.[1] For instance, in two dimensions, a hyperplane is a flat one-dimensional subspace—in other words, a line. In three dimensions, a hyperplane is a flat two-dimensional subspace—that is, a plane. In p > 3 dimensions, it can be hard to visualize a hyperplane, but the notion of a ( p − 1)-dimensional flat subspace still applies.

The mathematical definition of a hyperplane is quite simple. In two dimensions, a hyperplane is defined by the equation

\[\beta_0 + \beta_1 X_1 + \beta_2 X_2 = 0 \quad (9.1)\]

for parameters β 0 , β 1, and β 2. When we say that (9.1) “defines” the hyperplane, we mean that any X = ( X 1 , X 2) [T] for which (9.1) holds is a point on the hyperplane. Note that (9.1) is simply the equation of a line, since indeed in two dimensions a hyperplane is a line.

Equation 9.1 can be easily extended to the p -dimensional setting:

\[\beta_0 + \beta_1 X_1 + \dots + \beta_p X_p = 0 \quad (9.2)\]

defines a p -dimensional hyperplane, again in the sense that if a point X = ( X 1 , X 2 , . . . , Xp ) [T] in p -dimensional space (i.e. a vector of length p ) satisfies (9.2), then X lies on the hyperplane.

Now, suppose that X does not satisfy (9.2); rather,

\[\beta_0 + \beta_1 X_1 + \dots + \beta_p X_p > 0\]

Then this tells us that X lies to one side of the hyperplane. On the other hand, if

\[\beta_0 + \beta_1 X_1 + \dots + \beta_p X_p < 0\]

then X lies on the other side of the hyperplane. So we can think of the hyperplane as dividing p -dimensional space into two halves. One can easily determine on which side of the hyperplane a point lies by simply calculating the sign of the left-hand side of (9.2). A hyperplane in two-dimensional space is shown in Figure 9.1.