
4.2: Hyperplanes

Vectors in $$\mathbb{R}^{n}$$ can be hard to visualize. However, familiar objects like lines and planes still make sense: The line $$L$$ along the direction defined by a vector $$v$$ and through a point $$P$$ labeled by a vector $$u$$ can be written as $$L=\{ u + tv | t \in \mathbb{R} \}\, .$$

Sometimes, since we know that a point $$P$$ corresponds to a vector, we will be lazy and just write $$L=\{P + tv | t \in \mathbb{R} \}$$.

Example 44

$$\left\{ \begin{pmatrix}1\\2\\3\\4\end{pmatrix} + t\begin{pmatrix}1\\0\\0\\0\end{pmatrix} \middle\arrowvert t \in \mathbb{R} \right\}$$ describes a line in $$\mathbb{R}^{4}$$ parallel to the $$x_1$$-axis.

Given two non-zero vectors $$u,v$$, they will $$\textit{usually}$$ determine a plane,

unless both vectors are in the same line, in which case, one of the vectors is a scalar multiple of the other. The sum of $$u$$ and $$v$$ corresponds to laying the two vectors head-to-tail and drawing the connecting vector. If $$u$$ and $$v$$ determine a plane, then their sum lies in the plane determined by $$u$$ and $$v$$.

The plane determined by two vectors $$u$$ and $$v$$ can be written as $$\{ P + su + tv | s, t \in \mathbb{R} \}\, .$$

Example 45

$$\left\{ \begin{pmatrix}3\\1\\4\\1\\5\\9\end{pmatrix} + s\begin{pmatrix}1\\0\\0\\0\\0\\0\end{pmatrix} + t\begin{pmatrix}0\\1\\0\\0\\0\\0\end{pmatrix} \middle\arrowvert s, t \in \mathbb{R} \right\}$$ describes a plane in 6-dimensional space parallel to the $$xy$$-plane.

We can generalize the notion of a plane with the following recursive definition. (That is, infinitely many things are defined in the following line.)

Definition

A set of $$k$$ vectors $$v_{1}, \ldots, v_{k}$$ in $$\mathbb{R}^{n}$$ with $$k\leq n$$ determines a $$k$$-dimensional $$\textit{hyperplane}$$, unless any of the vectors $$v_{i}$$ lives in the same hyperplane determined by the other vectors. If the vectors do determine a $$k$$-dimensional hyperplane, then any point in the hyperplane can be written as:

$\left\{ P + \sum_{i=1}^{k} \lambda_{i}v_{i}\, |\, \lambda_{i} \in \mathbb{R} \right\}$

When the dimension $$k$$ is not specified, one usually assumes that $$k=n-1$$ for a hyperplane inside $$\mathbb{R}^{n}$$.