Let\(B = \left\{ {{{\bf{b}}_{\bf{1}}},{{\bf{b}}_{\bf{2}}},{{\bf{b}}_{\bf{3}}}} \right\}\) and \(D = \left\{ {{{\bf{d}}_{\bf{1}}},{{\bf{d}}_{\bf{2}}}} \right\}\) be bases for vector space \(V\) and \(W\), respectively. Let \(T:V \to W\) be a linear transformation with the property that

\(T\left( {{{\bf{b}}_1}} \right) = 3{{\bf{d}}_1} - 5{{\bf{d}}_2}\), \(T\left( {{{\bf{b}}_2}} \right) = - {{\bf{d}}_1} + 6{{\bf{d}}_2}\), \(T\left( {{{\bf{b}}_3}} \right) = 4{{\bf{d}}_2}\)

Find the matrix for \(T\) relative to \(B\), and\(D\).

A matrix associated with a linear transformation \(T\) for \(V\) and \(W\)is given by \({\left( {T\left( {\bf{x}} \right)} \right)_C} = \left( {\begin{aligned}{{{\left( {T\left( {{{\bf{b}}_1}} \right)} \right)}_C}}&{{{\left( {T\left( {{{\bf{b}}_2}} \right)} \right)}_C}}& \cdots &{{{\left( {T\left( {{{\bf{b}}_n}} \right)} \right)}_C}}\end{aligned}} \right)\), where \(V\) and \(W\) are \(n\) and \(m\)-dimensional subspaces respectively, and \(B\), and \(C\) are the bases for \(V\), and \(W\).

It is given that \(T\left( {{{\bf{b}}_1}} \right) = 3{{\bf{d}}_1} - 5{{\bf{d}}_2}\), \(T\left( {{{\bf{b}}_2}} \right) = - {{\bf{d}}_1} + 6{{\bf{d}}_2}\) and \(T\left( {{{\bf{b}}_3}} \right) = 4{{\bf{d}}_2}\).

Write them in the form of vectors as shown below:

\({\left( {T\left( {{{\bf{b}}_1}} \right)} \right)_D} = \left( {\begin{aligned}3\\{ - 5}\end{aligned}} \right)\)

\({\left( {T\left( {{{\bf{b}}_2}} \right)} \right)_D} = \left( {\begin{aligned}{ - 1}\\6\end{aligned}} \right)\)

\({\left( {T\left( {{{\bf{b}}_3}} \right)} \right)_D} = \left( {\begin{aligned}0\\4\end{aligned}} \right)\)

Form a matrix\(T\) for the obtained vectors by using the formula \({\left( {T\left( {\bf{x}} \right)} \right)_C} = \left( {\begin{aligned}{{{\left( {T\left( {{{\bf{b}}_1}} \right)} \right)}_C}}&{{{\left( {T\left( {{{\bf{b}}_2}} \right)} \right)}_C}}& \cdots &{{{\left( {T\left( {{{\bf{b}}_n}} \right)} \right)}_C}}\end{aligned}} \right)\), where \(n = 3\).

\(\begin{aligned}{c}{\left( {T\left( {\bf{x}} \right)} \right)_D} &= \left( {\begin{aligned}{{{\left( {T\left( {{{\bf{b}}_1}} \right)} \right)}_D}}&{{{\left( {T\left( {{{\bf{b}}_2}} \right)} \right)}_D}}&{{{\left( {T\left( {{{\bf{b}}_3}} \right)} \right)}_D}}\end{aligned}} \right)\\ &= \left( {\begin{aligned}3&{}&{ - 1}&{}&0\\{ - 5}&{}&6&{}&4\end{aligned}} \right)\end{aligned}\)

So, the required matrix is \(\left( {\begin{aligned}3&{}&{ - 1}&{}&0\\{ - 5}&{}&6&{}&4\end{aligned}} \right)\).