How do you see that $\theta x + (1-\theta)y$, $\theta \in \mathbb{R}$ is the straight line passing through $x,y$?

Question

A set $S$ is called affine if $ \theta x + (1-\theta)y \in S$, for all $x,y, \in S$, $\theta \in \mathbb{R}$

How do you see that $S$ contains all the straight lines passing through $x,y$?

It is easier to visualize for convex sets, because you can actually plug in the value of $\theta \in [0,1]$ to see that whatever the number you get must be contained in the line segment.

But here it is much more difficult to think about.

Answer 1

This is nothing more than the vector equation of a line: $$ s=\theta(x-y)+y $$

If $S$ contains all $s$ elements, then $S$ contain the straight line(s) passing through $x$ and $y$.