A fixed point of a mapping f is a point z0 with the property that f(z0) = z0.
(a) Does the linear mapping f(z) = az + b have a fixed point z0? If so, then find z0 in terms of a and b.
(b) Give an example of a complex linear mapping that has no fixed points.
(c) Give an example of a complex linear mapping that has more than one fixed point. [Hint: There is only one such mapping.]
(d) Prove that if z0 is a fixed point of the complex linear mapping f and if f commutes with the complex linear mapping g (see Problem 1), then z0 is a fixed point of g.
Problem 1
We say that two mappings f and g commute if f ◦ g(z) = g ◦ f(z) for all z. That is, two mappings commute if the order in which you compose them does not change the mapping.
(a) Can a nonidentity translation and a nonidentity rotation commute?
(b) Can a nonidentity translation and a nonidentity magnification commute?
(c) Can a nonidentity rotation and a nonidentity magnification commute?
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