5.2 .3 * Let $z$, $\omega$, and $\lambda$ be complex numbers, $|\lambda| \neq 1$, solve the equation for $z$, $\bar{z} - \lambda z = \omega$.


Taking the conjugate on both sides of the equation, we get $\bar{z}-\lambda z=\bar{\omega}$, which is $z-\bar{\lambda} \bar{z}=\bar{\omega}$. Multiplying both sides by $\lambda$ gives $\lambda z-|\lambda|^{2} \bar{z}=\lambda \bar{\omega}$ (2). Therefore, (1) + (2) yields $\bar{z}\left(1-|\lambda|^{2}\right)=\omega+\lambda \bar{\omega}$. Taking the conjugate again, we get $z(1-|\lambda|^{2})=\bar{\omega}+\bar{\lambda} \omega$. Since $|\lambda| \neq 1$, we have $z=\frac{\bar{\lambda} \omega+\bar{\omega}}{1-|\lambda|^{2}}$.