Question

Let $B=\left\{{\mathbf{b}}_{\mathbf{1}},{\mathbf{b}}_{\mathbf{2}}\right\}$, $C=\left\{{\mathbf{c}}_{\mathbf{1}},{\mathbf{c}}_{\mathbf{2}}\right\}$, and $D=\left\{{\mathbf{d}}_{\mathbf{1}},{\mathbf{d}}_{\mathbf{2}}\right\}$ be bases for a two-dimensional vector space.

a. Write an equation that relates the matrices $\underset{C\leftarrow B}{P}$, $\underset{D\leftarrow C}{P}$, and $\underset{D\leftarrow B}{P}$. Justify your result.

b. [M] use a matrix program either to help you find the equation or to check the equation you write. Work with three bases for ${\mathbb{R}}^{\mathbf{2}}$. (see Exercises 7-10)

Short answer

a. $\underset{D\leftarrow B}{P}=\underset{D\leftarrow C}{P}\underset{C\leftarrow B}{P}$

b. $\left[\begin{array}{cc}-3& 1\\ -5& 2\end{array}\right]$

Step-by-step solution

Find the stage matrix

The change of coordinate from B to C is

${\left[\mathbf{x}\right]}_C=\underset{C\to B}{P}{\left[\mathbf{x}\right]}_B$.

For a change of coordinate from C to D,

${\left[\mathbf{x}\right]}_D=\underset{D\to C}{P}{\left[\mathbf{x}\right]}_C$.

From both the equations,

${\left[\mathbf{x}\right]}_D=\underset{D\to C}{P}\underset{C\to B}{P}{\left[\mathbf{x}\right]}_B$.

The change of coordinate matrix from B to Dis

${\left[\mathbf{x}\right]}_D=\underset{B\to D}{P}{\left[\mathbf{x}\right]}_B$.

Therefore, for any vector ${\left[\mathbf{x}\right]}_B$ in ${\mathbb{R}}^2$,

$\underset{D\leftarrow B}{P}=\underset{D\leftarrow C}{P}\underset{C\leftarrow B}{P}$.

Find the change of coordinate matrix

The change of coordinate matrix from B to Cis

$\left[\begin{array}{cccc}{\mathbf{c}}_1& {\mathbf{c}}_2& {\mathbf{b}}_1& {\mathbf{b}}_2\end{array}\right]=\left[\begin{array}{cccc}1& -2& 7& -3\\ -5& 2& 5& -1\end{array}\right]$.

Use the following code in MATLAB to obtain the row-reduced echelon form:

$\begin{array}{l}>>\mathrm{A}=\left[\begin{array}{cccc}1& -2& 7& -3;\begin{array}{cccc}-5& 2& 5& -1\end{array}\end{array}\right];\\ >>\mathrm{U}=\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{f}\left(\mathrm{A}\right)\end{array}$

$\left[\begin{array}{cccc}1& -2& 7& -3\\ -5& 2& 5& -1\end{array}\right]\sim \left[\begin{array}{cccc}1& 0& -3& 1\\ 0& 1& -5& 2\end{array}\right]$

So, the change of coordinate matrix from B to Cis

$\underset{B\to C}{P}=\left[\begin{array}{cc}-3& 1\\ -5& 2\end{array}\right]$.