Find the image through $T$ of the square with vertices $1,-1,i,-i$

Question

$T:\mathbb{C}\rightarrow\mathbb{C} $ given by $ T(z)=(1+i)z$.

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$ T(1)=1+i $

$T(-1)=-1-i$

$T(i)=-1+i$

$T(-i)=1-i$

So the image is the square with vertices$ T(1),T(-1),T(i)$ and $(T-i)$ ?

Answer 1

Yes. Since $T$ is linear it is similarity transformation (preserves the angles and ratio of distances) with factor of similarity $\sqrt{2}$:

$$|T(z)-T(w)| = |(1+i)(z-w)|= |1+i||z-w| = \sqrt{2}|z-w|$$

so it takes square to square which is bigger than first one with factor $\sqrt{2}$.