Hyperplane

A hyperplane is a set of the form

$$\{x | a^T x = b\}$$

for some row vector $$a^T$$ and a scalar $$b$$. This set has several intuitive interpretations
 * It is the hyperplane with a normal vector $$A$$ and an offset distance of $$b$$ from the origin
 * $$\{x | a^T x = b\} = \{x | a^T (x-x_0)\} = x_0 + a^\perp$$ where $$x_0$$is any point belonging to the hyperspace, and $$a^\perp$$ is the set that contains all vectors that are orthogonal to $$a$$. Clearly, the latter sum operation is overloaded (its arguments are a set and a vector), and the definition of this overloading is straightforward (i.e., it's the set obtain by summing all the vectors in $$a^\perp$$ with the vector $$x_0$$).