We want to find a linear transformation that is equivalent to reflecting across the line $y = 2x$.  

1. Sketch the graph of $y = 2x$ and find the image of the point $\left(5, 0\right)$ when reflected across the line $y = 2x$.

   Hint (a):

   Move the red point so that it is reflected.
   Note, the segment needs to be perpendicular to the line.

2. Find a convenient point on the $y$-axis so that its image when reflected across the line will give integer values for the coordinate of the image. Find the image.

   Hint (b):

   Move the red point at $\left(0, 5\right)$ so that it is a reflection.

3. Use the points and images you found in parts (a) and (b) to find the linear transformation for reflecting across the line $y = 2x$.