Let $\vec{b}_1 = \left[\begin{array}{c} 1\cr -2 \end{array}\right]$ and $\vec{b}_2= \left[\begin{array}{c} 2\cr -5 \end{array}\right].$ The set $B= \left\lbrace \vec{b}_1, \vec{b}_2 \right\rbrace$ is a basis for ${\mathbb R}^2.$ Let $T: {\mathbb R}^2 \rightarrow {\mathbb R}^2$ be a linear transformation such that $T(\vec{b}_1) = 3 \vec{b}_1 + 3 \vec{b}_2$ and $T(\vec{b}_2) = 6 \vec{b}_1 + 7 \vec{b}_2.$

(a) The matrix of $T$ relative to the basis $B$ is
$[T]_B =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$ .

(b) The matrix of $T$ relative to the standard basis $E$ for ${\mathbb R}^2$ is
$[T]_E =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$