I'm reading about condition numbers on Wikipedia, and in the section on matrices it says that
$$
\left(\max_{e \neq 0} \frac {\left\Vert A^{-1}e\right\Vert }{\left\Vert e\right\Vert }\right)\cdot \left(\max_{b \neq 0} \frac {\left\Vert b\right\Vert }{\left\Vert A^{-1}b\right\Vert } \right) =
\left(\max_{e \neq 0} \frac {\left\Vert A^{-1}e\right\Vert }{\left\Vert e\right\Vert } \right)\cdot \left(\max_{x \neq 0} \frac {\left\Vert Ax\right\Vert }{\left\Vert x\right\Vert } \right).
$$
($A$ is a non-singular $n\times p$ matrix, and $b$ is a $p\times 1$ vector.)
As far as I can tell this means that
$$
\left(\max_{b \neq 0} \frac {\left\Vert b\right\Vert }{\left\Vert A^{-1}b\right\Vert } \right) =
\left(\max_{x \neq 0} \frac {\left\Vert Ax\right\Vert }{\left\Vert x\right\Vert } \right),
$$
which I can't figure out how to prove. It's probably some stunningly obvious property of norms, but I can't for the life of me figure out what it is. FWIW, at this point in the article, it has not been specified which norm is being used.
It doesn't matter which norm is used. If we make the substitution $b = Ax$, we have $$ \max_{b \neq 0} \frac{\|b\|}{\|A^{-1}b\|} = \max_{x \neq 0} \frac{\|(Ax)\|}{\|A^{-1}(Ax)\|} = \max_{x \neq 0} \frac{\|Ax\|}{\|x\|} $$