Real Hyperplanes

Why

We generalize the notion of a point in $\R $, a line in $\R ^2$ and a plane in $\R ^3$.

Definition

A hyperplane is a $(n-1)$-dimensional affine set in $\R ^n$.

Discussion

Since the $n-1$-dimensional subspaces are the orthogonal complements of the one-dimensional subspaces, they are the sets which can be specified by

\[ \Set*{x \in \R ^n}{x \perp b} + a \]

for $a, b \in \R ^n$. The hyperplanes are translates of these,

\[ \begin{aligned} \Set*{x \in \R ^n}{x \perp b} + a &= \Set*{x + a}{\ip{x,b} = 0} \\ &= \Set*{y}{\ip{y - a,b} = 0} = \Set*{y}{\ip{y,b} = \beta }, \\ \end{aligned} \]

where $\beta = \ip{a,b}$.

Characterization

$H \subset \R ^n$ is a hyperplane if and only if there exists $\beta \in \R $ and nonzero $b \in \R ^n$ so that

\[ H = \Set*{x \in \R ^n}{\ip{x,b} = \beta }. \]

$b$ and $\beta $ are unique up to a common nonzero multiple. For example, $b, \beta $ and $2b, 2\beta $ give the same hyperplane.

Any such vector $b$ is called a normal (or normal vector) to the hyperplane.

If $H$ is a hyperplane not containing the origin, then there is a unique unit vecor $u$ and $\alpha > 0$ so that

\[ H = \Set{x \in \R ^n}{\ip{x,u} = \alpha } \]