Concrete examples of affine transformations

Question

I would like to ask if somebody know concrete examples for the following two situations:

1. affine transformation (not linear) which preserves a compact set (possibly in a infinite dimensional vector space);
2. As (1) but without the fact that preserves a compact set.

Thank you very much for you help and sorry for my english :-) Have a nice day

Answer 1

1. Consider a compact set $S$ which is symmetric around a line $x = a$ in the plane, where $a \ne 0$. For instance, let $S$ be a disc whose centre is in the line. Then reflection in the line $x = a$ is affine but not linear, and it preserves $S$.

2. As suggested by Arthur, consider a translation.

Answer 2

Hint:

remember that any affine transformation is equivalent to a linear transformation followed by a translation. So a pure translation is affine but not linear.