Affine Hull and Affine Combination

The smallest affine set containing Ω, and linear combinations with coefficients summing to 1.

Affine Hull and Affine Combination

Let $X$ be a vector space and let $\Omega\subset X$ .

The affine hull of $\Omega$ is the intersection of all affine sets containing $\Omega$ :

\operatorname{aff}(\Omega):=\bigcap\{C\subset X\mid C\text{ is affine and }\Omega\subset C\}.

A vector $x\in X$ is an affine combination of $\omega_1,\dots,\omega_m\in X$ if

x=\sum_{i=1}^m \lambda_i\omega_i \quad\text{with}\quad \sum_{i=1}^m \lambda_i=1.

Affine combinations are the natural building blocks of affine sets, just as convex combinations are for convex sets.

Examples:

If $m=2$ and $\lambda_1=\lambda$ , $\lambda_2=1-\lambda$ , then $x=\lambda\omega_1+(1-\lambda)\omega_2$ parameterizes the line through $\omega_1,\omega_2$ as $\lambda$ varies over $\mathbb{R}$ .
In $\mathbb{R}^n$ , $\operatorname{aff}(\Omega)$ is the smallest affine subspace containing $\Omega$ .