Find the linear maps having form w = f(z) = Az + B that map the unit square with

Question

Find the linear maps having form w = f(z) = Az + B that map the unit square with corners at {0, 1, 1 + i, i} to an image square with the following properties: - the image area is four times larger; - the image sides are rotated pi/4 counter-clockwise (positive sense); - the image square is centred at 2i. Explain how to find a pair of complex-valued constants. A, B, that performs this map. Be clear about your method, as there is more than one such linear map. List all such linear maps. (Hint: try constructing a sequence of maps from the pre-image square to the final image square.)

Explanation / Answer

First of all because of the linear relationship leads to the conclusion that the shape does not change after the mapping into the (w)-plane. let (A = a + bi) and (B = c + di ) - image square is centered at (2i) ==> (w = 2i = Az+ B) where (z = center of the square in z-plane = 0.5 + 0.5 i) (0.5a - 0.5b + c = 0) ------(1) (0.5a + 0.5b + d = 2) ------(2) - image square is rotated by an angle of ( heta=45) degree ==> (Im(w)/Re(w) = 1) for z = real. say 1 or (rac{b + d}{a + c} = 1) -------(3) - image square's area is 4 times larger than original square. In w-plane, ((side)^2 = 4) ==> ((a + c)^2 + (b + d)^2 = 4 ) -------(4) [from the image of side on x-axis in z-plane] from eq. (3) and (4) (a+c = b+d = pm sqrt{2} ) -------(5) from eq. (1) + (2) a+c+d = 2 with eq.(5) (d = 2 pm sqrt{2}) --------(6) (b = -2 ) from eq.(2) - (1) (b - c + d = 2 ) ==> (c = 2 mp sqrt{2} ) with eq.(5) (a = -2 pm 2sqrt{2} ) finally, there are two sets of a,b,c,d. -------------------------------------------------- First (a = -2 + 2sqrt{2} ) (b = -2 ) (c = 2 - sqrt{2} ) (d = 2 + sqrt{2}) and Second, (a = -2 - 2sqrt{2} ) (b = -2 ) (c = 2 + sqrt{2} ) (d = 2 - sqrt{2} )

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