Difficult# Compositions of Transformations of the Plane

LINALG-ENXKQU

Let $T_1$, $T_2$, and $T_3$ be linear transformations $\mathbb{R}^2\to\mathbb{R}^2$ whose composition $T_1\circ T_2\circ T_3$ is the transformation given by vertical scaling by a factor $\cfrac{1}{3}$.

Suppose $T_3$ is (counter-clockwise) rotation by $\cfrac{\pi}{2}$ and $T_2$ is horizontal scaling by a factor of $\cfrac{1}{3}$.

What is $T_1$?