SECTION $3\mathrm{.}5$ EXERCISES

For each of the following, find the transition matrix corresponding to the change of basis from $\{u_1,u_2\}$ to $\{e_1,e_2\}.$

$($a$)u_1=(1,1{)}^T,u_2=(-1,1{)}^T$

$($b$)u_1=(1,2{)}^T,u_2=(2,5{)}^T$

$($c$)u_1=(0,1{)}^T,u_2=(1,0{)}^T$

Let $u_1=(1,1,1{)}^T,u_2=(1,2,2{)}^T,$ and $u_3=(2,3,4{)}^T.$

$($a$)$ Find the transition matrix corresponding to the change of basis from $\{e_1,e_2,e_3\}$ to $\{u_1,u_2,u_3\}.$

$($b$)$ Find the coordinates of each of the following vectors with respect to the ordered basis $\{u_1,u_2,u_3\}.$

$($i$)(3,2,5{)}^T$

$($ii$)(1,1,2{)}^T$

$($iii$)(2,3,2{)}^T$