3.4f Hyperplanes

Hyperplane

A hyperplane in an $n$-dimensional vector space ${\mathcal{X}}^n$ is a flat, affine subspace of dimension $n-1$ that divides ${\mathcal{X}}^n$ into two half-spaces; one below the hyperplane ${\mathcal{H}}_{-}$ and one above the hyperplane ${\mathcal{H}}_{++}$. In $n$-dimensional Euclidean space, It is defined as the set of all points $\vec{x}\in {\mathbb{R}}^n$ that satisfy a linear equation:
$\mathcal{H}=\{x\in \mathbb{R}:a^{T}x=b\},$
where $a\in {\mathbb{R}}^n$ ($\vec{a}\ne 0$) and $b\in \mathbb{R}$ are given.