Support Vector Machines — Lecture series — Equation of a Hyperplane

In the previous post, we spoke about what hyperplanes are. In this post, we will be looking at the equation of a hyperplane and how to derive it.

Learning objective:

Understand the equation of a hyperplane.

Main question:

How do you mathematically represent an n-dimensional plane in an n+1- dimensional space?

Well to help us answer this question, let’s start from the following likely instance: how do we mathematically represent a line that lies within a 3 dimensional space?

We know from basic geometry that the equation of a line is y = ax + b

Note that ‘y’ and ‘x’ in the equation above are just variables and they can be changed to any other variables such as ‘x1’ and ‘x2’. So let’s change these variables to get the representation of the equation of a line as:

x2 = ax1 + b

Let’s then move everything over to the left side of the equation to obtain the following:

-ax1 + x2 — b = 0

Then multiply through by -1 to get:

Equation 1: ax1 — x2 + b = 0

If we define two dimensional vectors x= (x1, x2) and w = (a, -1), we obtain another notation for the equation of the line represented in Equation 1:

Equation 2: w . x +b = 0

Since in equation 2 we are using vectors, it indicates that we can accommodate and represent higher dimensions. Hence equation 2 represents the equation of a hyperplane.