$)$ Let f:R$2$R$2$f:R$2$R$2$ be the linear transformation defined by

f$($x$)=[2342]$x$.$f$($x$)=[2432]$x$.$

Let

BC$==\{1,2,1,3\},\{1,2,1,3\},$B$=\{1,2,1,3\},$C$=\{1,2,1,3\},$

be two different bases forR$2$R$2.$

Find the matrix $[$f$]$BB$[$f$]$BB for ff relative to the basis BB$.$

$[$f$]$BB$=[$f$]$BB$=[]$

Find the matrix $[$f$]$CC$[$f$]$CC for ff relative to the basis CC$.$

$[$f$]$CC$=[$f$]$CC$=[]$

Find the transition matrix $[$I$]$BC$[$I$]$CB from CC to BB$.$

$[$I$]$BC$=[$I$]$CB$=[]$

Find the transition matrix $[$I$]$CB$[$I$]$BC from BB to CC$.($Note: $[$I$]$CB$=([$I$]$BC$)1[$I$]$BC$=([$I$]$CB$)1.)$

$[$I$]$CB$=[$I$]$BC$=[]$

On paper, check that $[$I$]$CB$[$f$]$BB$[$I$]$BC$=[$f$]$CC$[$I$]$BC$[$f$]$BB$[$I$]$CB$=[$f$]$CC$.$