Hyperplane

A hyperplane is a set of the form \(\{x \mid a^Tx=b\}\) such that \(a \in R^n\) and \(a \neq 0\) and \(b \in R\). In \(R^2\) the hyperplane is a line. In \(R^3\) the hyperplane is a plane.

Hyperplanes are not vector spaces unless they pass through the origin. A hyperplane also divides a vector space \(R^n\) into two [Halfspace] .

A hyperplanes is [Convex Set].

Proof:

Let \(x_1\) and \(x_2\) be two points in the hyperplane. Then \(a^Tx_1=b\) and \(a^Tx_2=b\). Now let \(c = \theta x_1 + (1-\theta) x_2\). For \(c\) to lie in the hyperplane, the condition \(a^Tc=b\) must be satisfied.

Now

\[\begin{align*} a^Tc &= a^T(\theta x_1 + (1-\theta) x_2)\\ &= \theta a^T x_1 + (1-\theta) a^T x_2 \\ &= \theta b + (1-\theta) b \\ &= \theta b + b - \theta b \\ &= b \\ \implies a^Tc &= b \end{align*}\]

Since the condition \(a^Tc=b\) is satisfied, we can say that the [Convex Combination] of any two points in the hyperplane also lies in the hyperplane. Hence we prove that hyperplanes are convex sets.