Exercise 11.11.1

Which of the following linear transformations $T:\mathbb{C}^{2}\to\mathbb{C}^{2}$ is unitary?

1. $T\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\left[\begin{array}{c}x\\ iy\end{array}\right]$.

2. $T\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\left[\begin{array}{c}iy\\ -ix\end{array}\right]$.

3. $T\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\left[\begin{array}{c}x+iy\\ ix+y\\ \end{array}\right]$.

4. $T\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\left[\begin{array}{c}\frac{1}{\sqrt{2}}x+\frac{i}{\sqrt{2}}y\\ \frac{1}{\sqrt{2}}x-\frac{i}{\sqrt{2}}y\\ \end{array}\right]$.