Complex Analysis - Maths KU

2.3 Linear Mappings 77
In Problems 17–20, find a linear mapping that maps the set S onto the set S ′ . (Note:
there may be more than one linear mapping that works.)
17. S is the triangle with vertices 0, 1, and 1 + i. S ′ is the triangle with vertices
2i, 3i, and −1+3i.
18. S is the circle |z − 1| =3. S ′ is the circle |z + i| =5.
19. S is the imaginary axis. S ′ is the line through the points i and 1 + 2i.
20. S is the square with vertices 1 + i, −1+i, −1 − i, and 1 − i. S ′ is the square
with vertices 1, 2 + i, 1+2i, and i.
21. Find two different linear mappings that map the square with vertices 0, 1, 1+i,
and i, onto the square with vertices −1, 0, i, −1+i.
22. Find two different linear mappings that map the half-plane Re(z) ≥ 2ontothe
half-plane Re(z) ≥ 5.
23. Consider the line segment parametrized by z(t) =z0 (1 − t)+z1t, 0≤ t ≤ 1.
(a) Find a parametrization of the image of the line segment under the translation
T (z) =z + b, b �= 0. Describe the image in words.
(b) Find a parametrization of the image of the line segment under the rotation
R(z) =az, |a| = 1. Describe the image in words.
(c) Find a parametrization of the image of the line segment under the magnification
M(z) =az, a>0. Describe the image in words.
24. Repeat Problem 23 for the circle parametrized by z(t) =z0 + re it .
25. In parts (a)–(c), express the given composition of mappings as a linear mapping
f(z) =az + b.
(a) rotation through π/4, magnification by 2, and translation by 1 + i
(b) magnification by 2, translation by √ 2, and rotation through π/4
√
2, rotation through π/4, then magnification by 2
(c) translation by 1
2
(d) What do you notice about the linear mappings in (a)–(c)?
26. Consider the complex linear mapping f(z) = � 1+ √ 3i � z + i. In each part,
find the translation T , rotation R, and magnification M that satisfy the given
equation and then describe the mapping f in words using T , R, and M.
(a) f(z) =T ◦ M ◦ R(z). (b) f(z) =M ◦ T ◦ R(z)
(c) f(z) =R ◦ M ◦ T (z)
Focus on Concepts
27. (a) Prove that the composition of two translations T1(z) =z + b1, b1 �= 0, and
T2(z) =z + b2, b2 �= 0, is a translation. Does the order of composition
matter?
(b) Prove that the composition of two rotations R1(z) =a1z, |a1| = 1, and
R2(z) =a2z, |a2| = 1, is a rotation. Does the order of composition matter?
(c) Prove that the composition of two magnifications M1(z) =a1z, a1 > 0, and
M2(z) =a2z, a2 > 0, is a magnification. Does the order of composition
matter?