9.1 Hyperplane

image credit: Deep AI

• A hyperplane is a \(p-1\)-dimensional flat subspace of a \(p\)-dimensional space. For example, in a 2-dimensional space, a hyperplane is a flat one-dimensional space: a line.
• (standard form) Definition of 2D hyperplane in 3D space: \[\beta_{0} + \beta_{1}X_{1} + \beta_{2}X_{2} + \beta_{3}X_{3}= 0\]
• (inner products) Any \(X\) s.t. \(X = (X_{1}, X_{2})^T\) for which the equation above is satisfied is a point on the hyperplane.

Additional resource: Deep AI