4.2: Hyperplanes

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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 $L=\{P + tv | t \in \mathbb{R} \}$.

$\left\{ \begin{pmatrix}1\\2\\3\\4\end{pmatrix} + t\begin{pmatrix}1\\0\\0\\0\end{pmatrix} \middle\arrowvert t \in \mathbb{R} \right\}$ 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,

$uvplane.jpg$

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 $v$.

$u+v.jpg$

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

\[\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 $\mathbb{R}^{n}$.

Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)

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