Householder reduction of an overdetermined consistent system
= Householder reduction of an overdetermined consistent system
{title2=$Q^Tb=(c,0)^T$}
For full column rank $A\in\mathbb R^{m\times n}$, a full <QR factorization> gives $Q^TA=(R_1,0)^T$ with invertible upper triangular $R_1$. The equation $Ax=b$ is consistent exactly when the final $m-n$ entries of $Q^Tb$ vanish. If consistent, <back substitution> gives its unique solution. This separates consistency from uniqueness in overdetermined systems.