Adjoint operator (10 min)

Consider the complex two-dimensional Hilbert space with basis vectors $(1,0)$ and $(0,1)$. Use the definition of the adjoint operator to prove the following for the adjoint $A^{\dagger}$ of the operator $A$: If $A$ is given as a complex two-by- two matrix,

$$A = \left( \begin{array}{cc} a & b\\ c & d \end{array}\right) \... ...dagger} = \left( \begin{array}{cc} a^* & c^*\\ b^* & d^* \end{array}\right).$$