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

[Juwane]

Homework Statement

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

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 $$w=f(z)=\frac{3+2i}{13}z + \frac{7+9i}{13}$$

Homework Equations

Maybe these would help:

$$x = \frac{z + \overline{z} }{2}$$ and $$y = \frac{z - \overline{z} }{2i}$$

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 $$z_{1} = 2$$ to get $$1+i$$. The answer is: $$-1+i$$ , but we can't add this to $$z_{2} = -3i$$, since that would give us $$1-2i$$ whereas we must get 3. Is there any other way to find out?

 

[Mark44]

Homework Statement

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

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 $$w=f(z)=\frac{3+2i}{13}z + \frac{7+9i}{13}$$

Homework Equations

Maybe these would help:

$$x = \frac{z + \overline{z} }{2}$$ and $$y = \frac{z - \overline{z} }{2i}$$

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 $$z_{1} = 2$$ to get $$1+i$$. The answer is: $$-1+i$$ , but we can't add this to $$z_{2} = -3i$$, since that would give us $$1-2i$$ 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.

 

[Dick]

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.

 

[Mark44]

Dick is right. If w₂ is 3, then the formula given as an answer is wrong.

 

[Juwane]

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.