In Warping basics we saw how a basic warp/transformation can be
represented with a matrix. To make more complex warpings (for example to
define a translation, rotation and scale as one warp) the individual
matrices have to be multiplied through matrix multiplication. However
matrix multiplication is not commutative, so the order of the set of
matrices you use for the multiplication is going to be very important.

The first warping should be placed as the left-most matrix. The second
warping to the right of that and so on. The second transformation is
going to occur on the warped coordinates of the first. As an example
for merging a few transforms into one matrix, the multiplication below
represents the rotation of an image about a point
\([\matrix{U&V}]\) anticlockwise from the horizontal axis by an
angle of \(\theta\). To do this, first we take the origin to
\([\matrix{U&V}]\) through translation. Then we rotate the image,
then we translate it back to where it was initially. These three
operations can be merged in one operation by calculating the matrix
multiplication below: