Thin QR and a least-squares fit

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Problem

Let $A$ have columns $a_1=(1,0,0)^{\mathsf T}$ and $a_2=(1,1,0)^{\mathsf T}$ , and let $b=(1,2,3)^{\mathsf T}$ . Find a thin QR factorization $A=QR$ , use it to solve $\min_x\|Ax-b\|_2$ , and give the fitted vector and residual.

Reveal answer or reference solution

Gram-Schmidt gives $q_1=(1,0,0)^{\mathsf T}$ and $q_2=(0,1,0)^{\mathsf T}$ , so

Q=\begin{bmatrix}1&0\\0&1\\0&0\end{bmatrix},\qquad R=\begin{bmatrix}1&1\\0&1\end{bmatrix}

The least-squares equation is $Rx=Q^{\mathsf T}b=(1,2)^{\mathsf T}$ . Thus $x_2=2$ and $x_1=-1$ . The fitted vector and residual are

Ax=(1,2,0)^{\mathsf T},\qquad b-Ax=(0,0,3)^{\mathsf T}

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Thin QR and a least-squares fit

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