  Let $K$ be a field, and let $\Aut \,K^2$ be the group of polynomial
automorphisms of $K^2$. We investigate which subgroups are linear or not. In
characteristic zero, there are small nonlinear subgroups and some big linear
subgroups. When $K$ has finite characteristic, the whole group $\Aut\,K^2$ is
linear whenever $K$ is finite, and nonlinear otherwise.
