Question

Let\(D = \left\{ {{{\bf{d}}_1},{{\bf{d}}_2}} \right\}\) and \(B = \left\{ {{{\bf{b}}_1},{{\bf{b}}_2}} \right\}\) be bases for vector space \(V\) and \(W\), respectively. Let \(T:V \to W\) be a linear transformation with the property that

\(T\left( {{{\bf{d}}_1}} \right) = 2{{\bf{b}}_1} - 3{{\bf{b}}_2}\), \(T\left( {{{\bf{d}}_2}} \right) = - 4{{\bf{b}}_1} + 5{{\bf{b}}_2}\)

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

Short answer

The matrix for \(T\) relative to \(D\) and \(B\) is \(\left( {\begin{aligned}2&{}&{ - 4}\\{ - 3}&{}&5\end{aligned}} \right)\).

Step-by-step solution

The matrix for a linear transformation 

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 \(W\).

Find the matrix for a linear transformation 

It is given that \(T\left( {{{\bf{d}}_1}} \right) = 2{{\bf{b}}_1} - 3{{\bf{b}}_2}\), and \(T\left( {{{\bf{d}}_2}} \right) = - 4{{\bf{b}}_1} + 5{{\bf{b}}_2}\).

Write them in the form of vectors as shown below:

\({\left( {T\left( {{{\bf{d}}_1}} \right)} \right)_B} = \left( {\begin{aligned}2\\{ - 3}\end{aligned}} \right)\)

\({\left( {T\left( {{{\bf{d}}_2}} \right)} \right)_B} = \left( {\begin{aligned}{ - 4}\\5\end{aligned}} \right)\)

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 \(n = 2\).

\(\begin{aligned}{\left( {T\left( {\bf{x}} \right)} \right)_B} &= \left( {\begin{aligned}{{{\left( {T\left( {{{\bf{d}}_1}} \right)} \right)}_B}}&{{{\left( {T\left( {{{\bf{d}}_2}} \right)} \right)}_B}}\end{aligned}} \right)\\ &= \left( {\begin{aligned}2&{}&{ - 4}\\{ - 3}&{}&5\end{aligned}} \right)\end{aligned}\)

So, the requiredmatrix is \(\left( {\begin{aligned}2&{}&{ - 4}\\{ - 3}&{}&5\end{aligned}} \right)\).