Notice that once you have taken $\displaystyle T(\lambda_1, \lambda_2, \lambda_3, \cdot\cdot\cdot)= (\lambda_2, \lambda_3, \lambda_4, \cdot\cdot\cdot)$ you have lost all information on $\displaystyle \lambda_1$. There is no way to know what $\displaystyle \lambda_1$ is in order to regain it. A more technical way of saying that is that T is not "one to one". Many different vectors $\displaystyle (\lambda_1, \lambda_2, \lambda_3, \cdot\cdot\cdot)$ will be mapped into the same $\displaystyle (\lambda_2, \lambda_3, \lambda_4, \cdot\cdot\cdot)$.
As for the second problem, the only difficulty is that there are an infinite number of correct answer- just choose one.