Do we deal with more complicated nonlinear transformations frequently in graphics? If so, would it ever be reasonable to somehow approximate one as linear due to the convenience or is that something that wouldn't make sense or work in practice?

With these decompositions, (considering homogenous coordinates), is there a decomposition that will allow us to split up the transformation into all of its possible components, or will you have to group some of them together?

Can we say that the spatial transformation is like the change of basis?