The Image of Extreme Subsets under Linear Maps

Question

Lax provides the following defintion for extreme subsets:

A subset $E$ of a convex set $K$ is an extreme subset when

1. $E$ is convex and non empty

2. Whenever $x \in E$ is expressed as $\frac{y+z}{2}$, $y,z \in K$, then $y,z \in E$

He provides two exercises:

1. If $M$ is a linear map from $M:X \rightarrow U$ and $E \subset K \subset U$ where $K$ is convex and $E$ is an extreme subset. Then the inverse image of $E$ is either empty or an extreme subset of the inverse image of $K$.

2. Give an example of how the image of an extreme subset under a linear map need not be an extreme subset of the image.

For 1, I can show this -- but I'm having trouble with the "empty" part, why can the inverse image of an extreme subset be empty?

And 2. I'm having trouble producing an example.

Thoughts?