F
F
Initial position
Animation
Final position
As you can verify by using the above radio buttons, the blue F and the red F have the same initial and final transformations, namely: Obviously, the blue F animates as one would expect, with its scaling axes invariant and ony the scaling factors varying; while the red F goes through some erratic transformations.

The only difference between the blue F and the red F is the coordinate systems used to express their transformations. The blue F is expressed in the default coordinate system up to a translation, while the red F is expressed in a coordinate system rotated by 45°. That is achieved by using nested div's for the red F, carrying the 45° rotations. In this coordinate system, the red F's scaling of factor 5 along the native X axis is now expressed by this matrix:
1312
1213

However, geometrically, it is still exactly the same, as you can see by checking the radio buttons above.

The problem is that despite having the same parameters, and in particular the same initial and final transforms, the two animations look very different from each other. This is because the way that the CSS Transforms spec mandates matrix interpolation to be performed is not coordinate-independent, i.e. isn't an interpolation of just the geometric transformations but rather an interpolation of quantities that depend on the coordinate system, which this example hopes to show isn't what we should want to be interpolating here.

An example of a coordinate-independent decomposition of linear transformations is the polar decomposition. It has been used before to interpolate transformations in computer graphics, see e.g. this article.

Let's finish by quoting Einstein on how making a similar mistake (assuming that a particular coordinate system was geometrically meaningful) cost him several years of work: "Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning." ("Albert Einstein: Philosopher-Scientist", Paul A. Schilpp, 1949, p. 67)