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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: $\phi(x) = \lambda x$?

rehband
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    a scaling is a Homothetic transformation, so yes b has to be zero – aflous Jul 11 '14 at 14:51
  • If $\lambda\ne 1$, then $x\mapsto \lambda x+b$ has a unique fixed point and is a scaling about that point. I doubt there is agreement between all texts about whether a "scaling", without further qualification, is required to fix the origin in particular. – hmakholm left over Monica Jul 11 '14 at 15:06

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