> [!tldr] > **Affine independence** is a relationship between a set of functions $T_{i}:\mathcal{X} \to \mathbb{R}$, where they cannot be linearly combined to give a constant function (except the trivial solution). That is, $\begin{align*} \not\exists c_{1},\dots,c_{k+1} &\in \mathbb{R}\text{ not all zero}:\\ &\forall x \in \mathcal{X}, \sum_{i=1}^{k}c_{i}T_{i}(x)=c_{k+1} \end{align*}$This is stronger than linear independence, which only bars combinations that give $c_{k+1}\equiv 0$. However, this is still equivalent to saying that b$\{ \mathbf{1}_{\mathcal{X}},T_{1},\dots,T_{k} \}\text{ is linearly independent.}$