The unitary matrix that implements rotations on real two-dimensional vectors can be written as

\[\mathbb{M}=\left(\begin{array}{cc}\cos \theta & \sin \theta \tag{3.153}\\-\sin \theta & \cos \theta\end{array}\right)\]

where \(\theta\) is the real-valued rotation angle.

(a) Verify that this matrix is indeed unitary.

(b) As a unitary matrix, it can be expressed as the exponential of a Pauli matrix in the form

\[\begin{equation*}\mathbb{M}=e^{i \theta \sigma_{j}} \tag{3.154}\end{equation*}\]

for some Pauli matrix \(\sigma_{j}\) defined in Eq. (3.67). Which Pauli matrix is it? Be sure to show your justification.