$[10\mathrm{.}4]$ Consider the $2\backslash$times $2$ orthogonal matrices $$cs cs

F$=$ s c $,$J$=$s c $,$

where s $=$ sin$\backslash$theta and c $=$ cos$\backslash$theta for some $\backslash$theta $.$ The first matrix has detF $=1$ and is a reflector $($a Householder reflector of dimension $2).$ The second has detJ $=1$ and effects a rotation instead of reflection, called a Givens rotation.

$($a$)$ Describe exactly what geometric effects left$-$multiplications by F and J have on the plane R$2.$

$($b$)$ Describe an algorithm for QR factorisation that is analogous to Algorithm $10\mathrm{.}1($in book$)$ but based on

Givens rotations instead of Householder reflections