6. (25 points)

a. (6 points) Project the vector $\mathbf{b}$ onto the nullspace of $A$, where

$\mathbf{b} = \begin{bmatrix} -3 \\ -2 \\ 1 \\ 2 \\ 1 \end{bmatrix} \text{ and } A = \begin{bmatrix} 1 & 2 & 1 & 0 & 0 \\ 2 & 5 & 1 & 1 & 0 \\ 3 & 7 & 2 & 2 & -2 \\ 4 & 9 & 3 & -1 & 4 \end{bmatrix}.$

b. Orthogonal Bases.

(i) (8 points) Apply the Gram-Schmidt process (Requirement: you must process following the column order, i.e. first column first, then second column, etc. or you will get zero point) to obtain orthonormal vectors from the columns of

$A = \begin{bmatrix} 1 & 4 & -2 & 2 \\ 2 & 1 & 3 & 2 \\ -1 & 0 & -2 & 0 \\ 0 & 1 & -1 & 8 \end{bmatrix}$

(ii) (3 points) As we known, the orthonormal vectors obtained from (i) cannot span $\mathbb{R}^4$. We can add some additional orthonormal vectors to those obtained from (i) and the new set of orthonormal vectors will span $\mathbb{R}^4$, and what are the additional orthonormal vectors?