Reduced QR least-squares solution ($Rx=Q^{T}b$)

Let $A=QR$ with $Q^{T}Q=I$ and square invertible upper triangular $R$. The normal equations for linear least squares reduce from $R^{T}Rx=R^T{Q}^{T}b$ to the system in the title. It is solved by back substitution. The geometric identity $\Vert Ax-b{\Vert }_2^2=\Vert Rx-Q^{T}b{\Vert }_2^2+\Vert (I-Q{Q}^T)b{\Vert }_2^2$ proves existence and uniqueness of the minimizer. For tall $Q$, $Q{Q}^T$ is an orthogonal projection, not the identity on the entire observation space.