Properties of Affine Sets and Affine Hulls

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

Proposition (key properties):

1. $\Omega$ is affine iff it contains all affine combinations of its elements.

2. If $\Omega,\Omega_1,\Omega_2$ are affine, then $\Omega_1+\Omega_2$ and $\lambda\Omega$ (for any scalar $\lambda$) are affine.

3. If $B:X\to Y$ is an affine mapping and $\Omega\subset X$ is affine, then $B(\Omega)\subset Y$ is affine; if $\Theta\subset Y$ is affine, then $B^{-1}(\Theta)$ is affine.

4. The affine hull $\operatorname{aff}(\Omega)$ is the smallest affine set containing $\Omega$ and admits the representation

   $$\operatorname{aff}(\Omega)=\left\{\sum_{i=1}^m \lambda_i\omega_i \ \middle|\ \sum_{i=1}^m\lambda_i=1,\ \omega_i\in\Omega,\ m\in\mathbb{N}\right\}.$$

5. A set $\Omega$ is a linear subspace iff it is affine and contains $0$.

Context: These results parallel the corresponding facts for convex sets and convex hulls , with affine combinations replacing convex combinations.