Consider the matrix
\begin{bmatrix} 1 & -6 \\ 3 & 6 \\ 4 & 8 \\ 5 & 0 \\ 7 & 8 \end{bmatrix}.
(a) Let $\beta = \{\vec{q}_1, \vec{q}_2, \vec{q}_3, \vec{q}_4, \vec{q}_5\}$ be an orthonormal basis for $\mathbb{R}^5$, such that $\vec{q}_1, \vec{q}_2$
span the column space of A. Which of the four fundamental subspaces is spanned
by $\vec{q}_3, \vec{q}_4, \vec{q}_5$?
(b) Find $\vec{q}_1$ and $\vec{q}_2$.
(c) Find $\beta$. You should find $\vec{q}_3, \vec{q}_4$, and $\vec{q}_5$.
(d) Find Q and R in the QR-decomposition of A.
(e) Using your result in (4d), find the best solution to the equation $A\vec{x} = \vec{b}$, if $\vec{b} =$
$(-3, 7, 1, 0, 4)$.