Problem $1.$ Consider the $32$ matrix A and right$-$hand side $b$ :

$A=\left[\begin{array}{cc}1& 2\\ -{10}^{-7}& {10}^{-7}\\ 0& {10}^{-14}\end{array}\right],$ and $,b=\left[\begin{array}{c}0\mathrm{.}8642\\ 0\mathrm{.}1440\\ 3\mathrm{.}1020\end{array}\right]$

$($a$)$ Use Matlab to find a singular value decomposition of A and use that to find ${}_2(A).$

$($b$)$ Use Matlab to find $B=A^{T}A,$ a singular value decomposition of $B,$ and ${}_2(B).$

$($c$)$ Use Matlab and the matrix $B$ to solve the normal equations $Bx=A^{T}b.$ Call the solution xnorm.

$($d$)$ Use Matlab to find a $QR-$decomposition of $A.$ Use your QR decomposition to find the matrices tilde$(Q)$ and tilde$(R)$ in a reduced $QR-$decomposition. Find the singular value decomposition of tilde$(R)$ and then find ${}_2(tilde(R)).$

$($e$)$ Use Matlab to solve the equations tilde$(R)x=$tilde$(Q{)}^{T}b.$ Call the solution $xqr.$

$($f$)$ Compare xnorm with $xq.$ Why should they be equal in theory? In fact, they are not very close to each other. Why is it that they are not that close?

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