2-1. (15 points) Given

$A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & 0 \\ 0 & 1 & 0 \end{bmatrix} \text{ and } b = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}.$

If the Gram-Schmidt process is applied to determine an orthonormal basis for $R(A) = \{b \in \mathbb{R}^m \mid b = A_{mn}x\}$ and $QR$ factorization of $A$, then, after the first orthonormal vector $q_1$ and $r_{11}$ are computed, we have

$Q = [q_1\ q_2\ q_3] = \begin{bmatrix} 0.5\sqrt{2} & - & - \\ 0 & - & - \\ 0.5\sqrt{2} & - & - \\ 0 & - & - \end{bmatrix} \text{ and } R = \begin{bmatrix} r_{11} & r_{12} & r_{13} \\ 0 & r_{22} & r_{23} \\ 0 & 0 & r_{33} \end{bmatrix} = \begin{bmatrix} \sqrt{2} & - & - \\ 0 & - & - \\ 0 & - & - \end{bmatrix}.$

(a) (5 points) Finish above process and determine $q_2$ and $q_3$, and fill in the columns of $Q$.

(b) (5 points) Finish above process and determine $R$.

(c) (5 points) Use the $QR$ factorization to find the least squares solution of $Ax = b$.