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Complex analysis: Find a function that maps z1 and z2 onto w1 and w2

  • Thread starter Juwane
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  • #1
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Homework Statement



What is the function (linear transformation) that maps [tex]z_{1} = 2[/tex] and [tex]z_{2} = -3i[/tex] onto [tex]w_{1} = 1+i[/tex] and [tex]w_{2} = 3[/tex]?

I think it's asking for the function that if you put 2 in it, it should give 1+i, and if you put -3i in the same function, it should give 3.

The answer given at the back of the book is [tex]w=f(z)=\frac{3+2i}{13}z + \frac{7+9i}{13}[/tex]

Homework Equations



Maybe these would help:

[tex]x = \frac{z + \overline{z} }{2}[/tex] and [tex]y = \frac{z - \overline{z} }{2i}[/tex]

The Attempt at a Solution



I have no idea how to even start. The horrible book I am using doesn't give a clue. One possible way is to see what do we have to [tex]z_{1} = 2[/tex] to get [tex]1+i[/tex]. The answer is: [tex]-1+i[/tex] , but we can't add this to [tex]z_{2} = -3i[/tex], since that would give us [tex]1-2i[/tex] whereas we must get 3. Is there any other way to find out?

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
33,173
4,858

Homework Statement



What is the function (linear transformation) that maps [tex]z_{1} = 2[/tex] and [tex]z_{2} = -3i[/tex] onto [tex]w_{1} = 1+i[/tex] and [tex]w_{2} = 3[/tex]?

I think it's asking for the function that if you put 2 in it, it should give 1+i, and if you put -3i in the same function, it should give 3.

The answer given at the back of the book is [tex]w=f(z)=\frac{3+2i}{13}z + \frac{7+9i}{13}[/tex]

Homework Equations



Maybe these would help:

[tex]x = \frac{z + \overline{z} }{2}[/tex] and [tex]y = \frac{z - \overline{z} }{2i}[/tex]

The Attempt at a Solution



I have no idea how to even start. The horrible book I am using doesn't give a clue. One possible way is to see what do we have to [tex]z_{1} = 2[/tex] to get [tex]1+i[/tex]. The answer is: [tex]-1+i[/tex] , but we can't add this to [tex]z_{2} = -3i[/tex], since that would give us [tex]1-2i[/tex] whereas we must get 3. Is there any other way to find out?
The textbook's answer gives f(z) as a linear polynomial with complex coefficients, which is one way to represent this function. It gives the right results for the two given complex numbers.

Edit: Scratch part of what I said. The formula gives the right result for 2 + i, but not the right result for -3i.

Another way to approach this problem is to use linear transformations, and treat complex numbers as vectors in R2.

From the given information, T(2) = T(<2, 0>T) = <1, 1>T, and
T(-3i) = T(<0, -3>T) = <3, 0>T.

Using the properties of linear transformations, it's easy to find T(<1, 0>T) and T(<0, 1>T). That means that you can find T(a + bi) = T(<a, b>T) = aT(<1, 0>T) + bT(<0, 1>T), for any complex number a + bi.
 
Last edited:
  • #3
Dick
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Homework Helper
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I think you mean w2=1. Double check the book. A linear function is f(z)=a*z+b for complex constants a and b. Just put in the given values and solve the two simultaneous equations for a and b.
 
Last edited:
  • #4
33,173
4,858
Dick is right. If w2 is 3, then the formula given as an answer is wrong.
 
  • #5
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I think you mean w2=1. Double check the book. A linear function is f(z)=a*z+b for complex constants a and b. Just put in the given values and solve the two simultaneous equations for a and b.
Yes, you're right. w2 is 1, not 3. I'll try to use the method you've given here. Thanks.
 

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