Find a complex linear function f(z) = az + b that rotates the point z counterclockwise through an angle of θ radians about the point z0 in the complex plane. [Hint: By Problem 1, this mapping is uniquely determined by the images of two points.]
Problem 1
In this problem we show that a linear mapping is uniquely determined by the images of two points.
(a) Let f(z) = az + b be a complex linear function with a ≠ 0 and assume that f(z1) = w1 and f(z2) = w2 where z1 ≠ z2. Find two formulas that express a and b in terms of z1, z2, w1, and w2. Explain why these formulas imply that the linear mapping f is uniquely determined by the images of two points.
(b) Show that a linear function is not uniquely determined by the image of one point. That is, find two different linear functions f1 and f2 that agree at one point.
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