Worksheet: Möbius Transformations

In this worksheet, we will practice interpreting möbius transformation in the complex plane.

Q1:

Consider the MΓΆbius transformations 𝑇 ( 𝑧 ) = 𝑧 + 3 𝑖 𝑖 𝑧 βˆ’ 2  and 𝑇 ( 𝑧 ) = 𝑖 𝑧 + 1 2 𝑧  , where 𝑧 β‰  2 𝑖 or 0.

Write an expression for the composition 𝑇 ∘ 𝑇 ( 𝑧 )   .

  • A 𝑇 ∘ 𝑇 ( 𝑧 ) = 7 𝑖 𝑧 + 1 ( βˆ’ 4 + 𝑖 ) 𝑧 + 1  
  • B 𝑇 ∘ 𝑇 ( 𝑧 ) = 2 𝑖 𝑧 βˆ’ 5 2 𝑧 + 6 𝑖  
  • C 𝑇 ∘ 𝑇 ( 𝑧 ) = 7 𝑖 𝑧 + 1 βˆ’ 5 𝑧 + 𝑖  
  • D 𝑇 ∘ 𝑇 ( 𝑧 ) = ( 1 + 𝑖 ) 𝑧 βˆ’ 2 + 3 𝑖 2 𝑧 + 6 𝑖  
  • E 𝑇 ∘ 𝑇 ( 𝑧 ) = ( 1 + 𝑖 ) 𝑧 + 1 + 3 𝑖 ( 2 + 𝑖 ) 𝑧 βˆ’ 2  

Q2:

A transformation that maps the 𝑧 -plane to the 𝑀 -plane is defined by 𝑇 ∢ 𝑧 ↦ 1 𝑧 , where 𝑧 β‰  0 .

Find an equation for the image of | 𝑧 | = 2 under the transformation.

  • A 𝑀 = 1 2
  • B 𝑀 = 2
  • C | 𝑀 | = 1 2
  • D | 𝑀 | = 2
  • E | 𝑀 | = 1

Find an equation for the image of a r g ( 𝑧 ) = 3 πœ‹ 4 .

  • A a r g ( 𝑀 ) = 3 πœ‹ 4
  • B a r g ( 𝑀 ) = 4 3 πœ‹
  • C a r g ( 𝑀 ) = βˆ’ πœ‹ 4
  • D a r g ( 𝑀 ) = πœ‹ 4
  • E a r g ( 𝑀 ) = βˆ’ 3 πœ‹ 4

Find a Cartesian equation for the image of I m ( 𝑧 ) = 2 .

  • A 2 𝑒 + ο€Ό 2 𝑣 + 1 4  = 1 1 6  
  • B 𝑒 + ο€Ό 𝑣 + 1 4  = 1 1 6  
  • C 𝑒 + ο€Ό 𝑣 + 1 2  = 1 4  
  • D 𝑣 = 1 2
  • E 𝑣 = 0

Find a Cartesian equation for the image of | 𝑧 βˆ’ 𝑖 | = 1 2 .

  • A 𝑒 + ο€Ό 𝑣 + 4 3  = 4 9  
  • B 𝑒 + ο€Ό 𝑣 + 4 3  = 4 9  
  • C 𝑒 + 𝑣 = 0
  • D 𝑒 + ο€Ό 𝑣 + 4 3  = 0  
  • E 𝑒 + ο€Ό 𝑣 + 8 3  = 0  

Q3:

A transformation that maps the 𝑧 -plane to the 𝑀 -plane is defined by 𝑇 ∢ 𝑧 ↦ 1 3 𝑧 βˆ’ 6 𝑖 .

Find the Cartesian equation for the image of | 𝑧 + 2 | = 3 .

  • A ο€Ό 𝑒 βˆ’ 2 3  + ο€Ό 𝑣 βˆ’ 2 3  = 1  
  • B ο€Ό 𝑒 βˆ’ 2 3  + ο€Ό 𝑣 + 2 3  = 1  
  • C ο€Ό 𝑒 βˆ’ 2 3  + ο€Ό 𝑣 + 2 3  = 0  
  • D 𝑒 βˆ’ 4 3 𝑒 + 𝑣 βˆ’ 4 3 𝑣 βˆ’ 1 6 𝑒 𝑣 βˆ’ 1 9 = 0  
  • E ο€Ό 𝑒 βˆ’ 2 2 1  + ο€Ό 𝑣 + 2 2 1  = 5 1 4 7  

Find the Cartesian equation for the image of R e ( 𝑧 ) = 5 .

  • A ο€Ό 𝑒 βˆ’ 1 1 5  βˆ’ 𝑣 = 1 2 2 5  
  • B ο€Ό 𝑒 βˆ’ 1 1 5  + 𝑣 = 1 2 2 5  
  • C ο€Ό 𝑒 βˆ’ 1 3 0  βˆ’ 𝑣 = 1 9 0 0  
  • D ο€Ό 𝑒 βˆ’ 1 3 0  + 𝑣 = 1 9 0 0  
  • E 𝑒 + 𝑣 = 5  

Q4:

A transformation, 𝑇 , that maps the 𝑧 -plane to the 𝑀 -plane is given by 𝑇 ( 𝑧 ) = ( 2 + 𝑖 ) 𝑧 + 4 𝑧 βˆ’ 𝑖 , where 𝑧 β‰  𝑖 .

Find a Cartesian equation for the image of the imaginary axis under the transformation 𝑇 .

  • A 𝑣 = 4 βˆ’ 3 2 𝑒
  • B ( 𝑒 βˆ’ 1 ) + ο€Ό 𝑣 + 5 2  = 1 2  
  • C 𝑣 = 3 2 𝑒 + 4
  • D 𝑣 = 3 2 𝑒 βˆ’ 4
  • E ( 𝑒 βˆ’ 1 ) + ο€Ό 𝑣 + 5 2  = 1 3 4  

Hence, find the image of the region I m ( 𝑧 ) > 0 under the transformation 𝑇 .

  • A ( 𝑒 βˆ’ 1 ) + ο€Ό 𝑣 + 5 2  < 1 3 4  
  • B ( 𝑒 βˆ’ 1 ) + ο€Ό 𝑣 + 5 2  > 1 3 4  
  • C 𝑣 > 4 βˆ’ 3 2 𝑒
  • D 𝑣 < 4 βˆ’ 3 2 𝑒
  • E 𝑣 > 3 2 𝑒 βˆ’ 4

Q5:

Which of the following MΓΆbius transformations maps | 𝑧 | = 1 to the real axis?

  • A 𝑇 ( 𝑧 ) = 𝑧 βˆ’ 2 𝑧 + 2
  • B 𝑇 ( 𝑧 ) = 𝑖 𝑧 βˆ’ 2 𝑖 𝑧 + 2
  • C 𝑇 ( 𝑧 ) = 𝑖 𝑧 + 𝑖 𝑧 + 1
  • D 𝑇 ( 𝑧 ) = 𝑧 βˆ’ 1 𝑧 + 1
  • E 𝑇 ( 𝑧 ) = 𝑖 𝑧 βˆ’ 𝑖 𝑧 + 1

Q6:

Suppose 𝐿 ∢ ℝ β†’ ℝ is a linear transformation with 𝐿 ( 2 ) = 7 . What is the value of 𝐿 ( 1 ) ?

  • A1
  • B7
  • CThis depends on the definition of 𝐿 .
  • D 2 7
  • E 7 2

Q7:

Suppose 𝐿 ∢ ℝ β†’ ℝ is a linear transformation. What is the value of 𝐿 ( 0 ) ?

Q8:

Which of the following MΓΆbius transformations, 𝑇 , maps βˆ’ 2 to 0, 0 to βˆ’ 3 , and has l i m  β†’ ∞ 𝑇 ( 𝑧 ) = βˆ’ 3 𝑖 ?

  • A 𝑇 ( 𝑧 ) = 3 𝑧 + 6 𝑖 𝑧 βˆ’ 3
  • B 𝑇 ( 𝑧 ) = 𝑧 + 2 𝑖 βˆ’ 3 𝑖 𝑧 βˆ’ 6 𝑖
  • C 𝑇 ( 𝑧 ) = βˆ’ βˆ’ 𝑧 +          
  • D 𝑇 ( 𝑧 ) = 𝑖 𝑧 βˆ’ 3 3 𝑧 + 6
  • E 𝑇 ( 𝑧 ) = βˆ’ 3 𝑖 𝑧 βˆ’ 6 𝑖 𝑧 + 2 𝑖

Q9:

Find the equation for the image of a r g ( 𝑧 ) = πœ‹ 2 under the transformation 𝑀 = 𝑧 7 𝑧 βˆ’ 2 𝑖 , 𝑧 β‰  2 𝑖 7 , which maps the 𝑧 -plane to the 𝑀 -plane.

  • A a r g ( 𝑀 ) = πœ‹ 2
  • B a r g ( 𝑀 ) = 0
  • C a r g ( 𝑀 ) = 2 πœ‹ 3
  • D a r g ( 𝑀 ) = πœ‹ 6
  • E a r g ( 𝑀 ) = βˆ’ πœ‹ 2

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