Find the image of the region 0<x<1 under the transformation $\displaystyle T: z \to iz$.

Attempt: I write z=a+ib where 0<a<1 and b can be any real number. So the transformation takes these z and output i(a+ib)=-b+ia.

On the complex plane that makes an infinite horizontal region bounded above by i and below by 0?

While the original graph would be an infinite vertical region bounded on the right by 1 and on the left by 0.

Am I right? I'm quite rusty, I'm trying to end my vacations by starting maths back.

Thanks for any insight.