Hyperplane

Description:

• A hyperplane in ${\mathbb{R}}^n$ is a set of vectors of the form $H=\{x\in {\mathbb{R}}^n:a^{T}x=b\}$
  • where $a\in {\mathbb{R}}^n,a\ne 0$, and $b\in \mathbb{R}$ is given
• We can also think of hyperplanes as the level sets of Linear Function
• When $b=0$, the hyperplane is simply the set of points that are orthogonal to $a$
  • ie, $H$ is a $n-1$ dimensional Subspace
• The hyperplane then separates the whole vector space into 2 regions:
  • $H_{--}=\{x:a^{T}x\le b\},H_{++}=\{x:a^{T}x>b\}$
• These regions are called halfspace