TITLE: The difference between affine set and affine hull
QUESTION [8 upvotes]: According to the definition of affine hull and affine set.
$$aff [C] = [\theta_1x_1+...+\theta_nx_n|x_1,...x_n \in C, \theta_1+...+\theta_n=1] $$
The data in affine hull is also in affine set. And vice versus. So what's difference between them?

REPLY [5 votes]: Affine set, Affine hull, Convex set and Convex hull
Affine set is a set which contains every affine combinations of points in it. 
For example, for two points $x,y\in R^2 $, an affine set is the whole line passing through these two points. (Note: $\theta_i$ could be negative as long as $\theta_1 + \theta_2 =1$. If all $\theta_i \geq 0$, it is called a convex set and it is the line segment between two points instead of the whole line.)
For affine hull, convex hull might be easier to visualize first. Let's say $A= \{ x,y,z | x,y,z \in R^2 \} $. Convex hull of A is a plastic wrap around x,y,z containing every points within the wrap and on the border. The reason it is within the wrap is because coefficients $\theta_i \geq 0 $. affine hull allow $ \theta_i $ to be negative, so affine hull will be entire $R^2$ unless all three points lie on a line, in that case , affine hull will be the whole line passing through these three points.
Defintion of Affine hull for an arbitrary set A is the smallest affine set that can contains all points in A. Affine hull for an affine set is the affine set itself.  
Hope it helps !!