Consider the ordered bases $B=((-5,-1),(-4,-1))$ and $C=((1,3),(4,-2))$ for the vector space ${\mathbb R}^2$.
a. Find the transition matrix from $C$ to the standard ordered basis $E=((1,0),(0,1))$.
$T_C^E =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
b. Find the transition matrix from $B$ to $E$.
$T_B^E =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
c. Find the transition matrix from $E$ to $B$.
$T_E^B =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
d. Find the transition matrix from $C$ to $B$.
$T_C^B =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
e. Find the coordinates of $u= (1,-3)$ in the ordered basis $B$. Note that $[u]_B = T_E^B [u]_E$.
$[u]_B =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
f. Find the coordinates of $v$ in the ordered basis $B$ if the coordinate vector of $v$ in $C$ is $[v]_C = (2,1)$.
$[v]_B =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$