"Let w = a + bi be a complex number w of modulus 1. Let T be the transformation defined by T(z) = wz ("multiplication by w"). Write z = x + iy and write T(Z) = x' + iy'. Give a formula for x' and y' in terms of x,y,a,b."

How do I go about getting the formula? I understand almost everything up to that point.

Question

"Let w = a + bi  be a complex number w of modulus 1. Let T be the transformation defined by T(z) = wz ("multiplication by w"). Write z = x + iy and write T(Z) = x' + iy'. Give a formula for x' and y' in terms of x,y,a,b." 

How do I go about getting the formula? I understand almost everything up to that point.

terenzreignz · Answer

Oh ok... simple enough, but the wording isn't :D
w = a + bi
z = x + yi

simple enough.

so, what is T(z)?

T(z) = wz

But express this in terms of w = a+bi and z = x+yi
and then simplify.
Could you do that?

anonymous · Answer

Yes, I think so. ax-by+i(ay+bx). Is that correct?

terenzreignz · Answer

Quite right.

So we would be right in saying that
T(z) = [ax - by] + i[ay + bx]
just as T(z) = wz

yes?

Well then, that about wraps it up.

The tricky part here is that x' is simply the REAL PART of T(z)
and y' is the IMAGINARY part.

Of course, if the wording had been like that, the question might have been a dead giveaway.

Do you have your answer now? ^_^

anonymous · Answer

Yes, I do! Thank you so much!

terenzreignz · Answer

Good. Always helps to know that you know :D

terenzreignz · Answer

Now, just in hindsight, I don't really see the significance of the modulus of w being 1... hmmm...

anonymous · Answer

I think it's just there to either a) throw us off or b) to help with another part of the exercise