So, we can write an affinity $\phi$ as $$\phi(x) = Ax + b$$ for some linear transformation $A$ and vector $b$.
What exactly does it mean for an affinity to be a "scaling"? Is this a mapping of the form $\phi(x) = \lambda x + b$ for some scalar $\lambda$ or must the translation also be zero: $\phi(x) = \lambda x$?