Hyperplane¶

Definition¶

Let $V$ be a vector space of dimension $n\ge1$ $1$ .

A linear hyperplane of $V$ is a subspace $H$ of dimension $n-1$ $1$ .
An affine hyperplane of $V$ is a translate $\boldsymbol{p}+H=\{\boldsymbol{p}+\boldsymbol{h}\colon \boldsymbol{h}\in H\}$ of a linear hyperplane $H$ , where $\boldsymbol{p}\in V$ $V$ .

In $\mathbb{R}^n$ the affine hyperplanes are exactly the sets

\{\boldsymbol{x}\in\mathbb{R}^n\colon a_1x_1+\cdots+a_nx_n=d\}\quad \text{with }\boldsymbol{a}=(a_1,\ldots,a_n)\neq\boldsymbol{0},

and such a hyperplane is linear exactly when $d=0$ $0$ , that is, when it passes through the origin. Used without a qualifier in $\mathbb{R}^n$ , "hyperplane" usually means an affine hyperplane; some linear algebra texts reserve it for linear hyperplanes, so the qualifier is worth stating.

Interpretation¶

A hyperplane in $\mathbb{R}^n$ is the solution set of one nontrivial linear equation $a_1x_1+\cdots+a_nx_n=d$ $d$ . One equation removes one dimension, so a hyperplane has dimension $n-1$ $1$ ; it generalizes lines in $\mathbb{R}^2$ and planes in $\mathbb{R}^3$ .

In the plane, the line through the origin of all x with a dot x = 0, a normal vector a = (-1, 2) at a right angle to it, and a parallel shifted line where a dot x = c.

In $\mathbb{R}^2$ the linear hyperplane $H=\{\boldsymbol{x}\colon\boldsymbol{a}\cdot\boldsymbol{x}=0\}$ with $\boldsymbol{a}=(-1,2)$ is the line $-x_1+2x_2=0$ through the origin. The vector $\boldsymbol{a}$ is perpendicular to it: every vector along $H$ , such as $(2,1)$ , has $\boldsymbol{a}\cdot\boldsymbol{x}=0$ $0$ . Changing the right-hand side to $c\neq0$ shifts the line parallel to itself, giving an affine hyperplane (dashed). The dashed line lies on the side away from $\boldsymbol{a}$ , so there $c<0$ $0$ .

In $\mathbb{R}^2$ the hyperplane $a_1x_1+a_2x_2=d$ is a line with normal vector $\boldsymbol{n}=(a_1,a_2)$ $(a_{1}, a_{2})$ . Change $d$ to shift it; drag $\boldsymbol{x}$ to see which side it is on.

The dimension count is the fundamental theorem of linear maps: $H$ is the kernel of the linear map $\boldsymbol{x}\mapsto\boldsymbol{a}\cdot\boldsymbol{x}$ from $\mathbb{R}^n$ to $\mathbb{R}$ , whose range is all of $\mathbb{R}$ when $\boldsymbol{a}\neq\boldsymbol{0}$ $0$ , so $\dim H=n-1$ $1$ . The condition $\boldsymbol{a}\neq\boldsymbol{0}$ matters: for $\boldsymbol{a}=\boldsymbol{0}$ the equation reads $0=d$ and describes all of $\mathbb{R}^n$ or the empty set. The normal vector $\boldsymbol{a}$ fixes the direction of the hyperplane and $d$ its position.

The inequality $\boldsymbol{a}\cdot\boldsymbol{x}\le d$ describes a half-space, the hyperplane together with one side of it. The constraints of a linear program are such half-spaces and hyperplanes, so its feasible region is an intersection of them.

A common slip is to expect a line in $\mathbb{R}^3$ : there a hyperplane is a plane, and a line needs two independent equations.

Key Results¶

The vector $\boldsymbol{a}=(a_1,\ldots,a_n)$ is a normal vector of the hyperplane $a_1x_1+\cdots+a_nx_n=d$ $d$ : it is orthogonal to every difference $\boldsymbol{x}-\boldsymbol{y}$ of two of its points.
An affine hyperplane $\boldsymbol{a}\cdot \boldsymbol{x}=d$ divides $\mathbb{R}^n$ into the two closed half-spaces $\{\boldsymbol{x}\colon \boldsymbol{a}\cdot \boldsymbol{x}\le d\}$ and $\{\boldsymbol{x}\colon \boldsymbol{a}\cdot \boldsymbol{x}\ge d\}$ $d}$ , which meet exactly in the hyperplane.
The hyperplanes of $\mathbb{R}^2$ are the lines and those of $\mathbb{R}^3$ are the planes.

Examples¶

$x+y=1$ is an affine hyperplane of $\mathbb{R}^2$ that is not linear, since it misses the origin.
$x+y+z=0$ is a linear hyperplane of $\mathbb{R}^3$ .
The $xy$ -plane $\{(x,y,z)\colon z=0\}$ is a linear hyperplane of $\mathbb{R}^3$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 5
Math III: Analysis · Lecture 4

Keywords: affine hyperplane, linear hyperplane, vector hyperplane, half-space Symbols: element of, real numbers

Last updated: 22 October 2026 19:04