Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there exists a matrix in $G$ that maps $u$ to $v$. Finally suppose that that exists an invertible matrix in $G$. Are there any non-trivial examples of such a $G$?

Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there exists a matrix in $G$ that maps $u$ to $v$. Are there any non-trivial examples of such a $G$?

Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ in $\mathbb{R}^2$ there is a matrix in $G$ that maps $u$ to $v$. Are there any transitive semigroups besides the obvious ones (diagonal, lower triangular, and upper triangular semigroups)?

Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there exists a matrix in $G$ that maps $u$ to $v$. Finally suppose that that exists an invertible matrix in $G$. Are there any non-trivial examples of such a $G$?