Solving the latter equation gives us $\mathbf{u}$ and thus the needed transformation $\mathbf{P}$. We do first however need to compute the scalar $k$ by taking the scalar product of the last equation with its transpose and using the fact that $\mathbf{P}^2=\mathbf{I}$. We get then $$(\mathbf{Pv})^T\mathbf{Pv} = k^{2} = \mathbf{v}^T\mathbf{v}= |v|^2 = \sum_{i=2}^{n}a_{i1}^2,$$ which determines the constant $ k = \pm v$.