Question

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

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

Find the matrix for $T$ relative to $B$, and$D$.

Short answer

The matrix for $T$ relative to $B$ and $D$ is $\left(\begin{array}{rlrlr}3& & -1& & 0\\ -5& & 6& & 4\end{array}\right)$.

Step-by-step solution

Matrix for a linear transformation 

A matrix associated with a linear transformation $T$ for $V$ and $W$is given by ${\left(T\left(\mathbf{x}\right)\right)}_C=\left(\begin{array}{rlrl}{\left(T\left({\mathbf{b}}_1\right)\right)}_C& {\left(T\left({\mathbf{b}}_2\right)\right)}_C& \cdots & {\left(T\left({\mathbf{b}}_n\right)\right)}_C\end{array}\right)$, 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 $T\left({\mathbf{b}}_1\right)=3{\mathbf{d}}_1-5{\mathbf{d}}_2$, $T\left({\mathbf{b}}_2\right)=-{\mathbf{d}}_1+6{\mathbf{d}}_2$ and $T\left({\mathbf{b}}_3\right)=4{\mathbf{d}}_2$.

Write them in the form of vectors as shown below:

${\left(T\left({\mathbf{b}}_1\right)\right)}_D=\left(\begin{array}{r}3\\ -5\end{array}\right)$

${\left(T\left({\mathbf{b}}_2\right)\right)}_D=\left(\begin{array}{r}-1\\ 6\end{array}\right)$

${\left(T\left({\mathbf{b}}_3\right)\right)}_D=\left(\begin{array}{r}0\\ 4\end{array}\right)$

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

$\begin{array}{rl}c{\left(T\left(\mathbf{x}\right)\right)}_D& =\left(\begin{array}{rlr}{\left(T\left({\mathbf{b}}_1\right)\right)}_D& {\left(T\left({\mathbf{b}}_2\right)\right)}_D& {\left(T\left({\mathbf{b}}_3\right)\right)}_D\end{array}\right)\\ & =\left(\begin{array}{rlrlr}3& & -1& & 0\\ -5& & 6& & 4\end{array}\right)\end{array}$

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