The Augmentation Pipeline

A vision team augments a training image by pushing every pixel's coordinate vector through three transformations, in this order:

Applied	Move	Matrix
1st	Shear with $k = 1$	$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$
2nd	Scale $x$ by $2$ , $y$ by $3$	$\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$
3rd	Rotate $90^\circ$ counter-clockwise	$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$

One corner pixel of the image starts at $v = (4, 1)$ .

Follow that corner through all three moves, then report its distance from the origin once the pipeline has finished.

Round your answer to two decimal places.