Householder reduction of an overdetermined consistent system ($Q^{T}b=(c,0{)}^T$)

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