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

a. Write an equation that relates the matrices \(\mathop P\limits_{C \leftarrow B} \), \(\mathop P\limits_{D \leftarrow C} \), and \(\mathop P\limits_{D \leftarrow B} \). 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 \({\mathbb{R}^{\bf{2}}}\). (see Exercises 7-10)

The change of coordinate from B to C is

\({\left[ {\bf{x}} \right]_C} = \mathop P\limits_{C \to B} {\left[ {\bf{x}} \right]_B}\).

For a change of coordinate from C to D,

\({\left[ {\bf{x}} \right]_D} = \mathop P\limits_{D \to C} {\left[ {\bf{x}} \right]_C}\).

From both the equations,

\({\left[ {\bf{x}} \right]_D} = \mathop P\limits_{D \to C} \mathop P\limits_{C \to B} {\left[ {\bf{x}} \right]_B}\).

The change of coordinate matrix from B to Dis

\({\left[ {\bf{x}} \right]_D} = \mathop P\limits_{B \to D} {\left[ {\bf{x}} \right]_B}\).

Therefore, for any vector \({\left[ {\bf{x}} \right]_B}\) in \({\mathbb{R}^2}\),

\(\mathop P\limits_{D \leftarrow B} = \mathop P\limits_{D \leftarrow C} \mathop P\limits_{C \leftarrow B} \).

The change of coordinate matrix from B to Cis

\(\left[ {\begin{array}{*{20}{c}}{{{\bf{c}}_1}}&{{{\bf{c}}_2}}&{{{\bf{b}}_1}}&{{{\bf{b}}_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&7&{ - 3}\\{ - 5}&2&5&{ - 1}\end{array}} \right]\).

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

\(\begin{array}{l} > > {\rm{ A }} = {\rm{ }}\left[ {{\rm{ }}\begin{array}{*{20}{c}}1&{ - 2}&7&{ - 3;\,\,\begin{array}{*{20}{c}}{ - 5}&2&5&{ - 1}\end{array}}\end{array}} \right];\\ > > {\rm{ U}} = {\rm{rref}}\left( {\rm{A}} \right)\end{array}\)

\(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&7&{ - 3}\\{ - 5}&2&5&{ - 1}\end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}1&0&{ - 3}&1\\0&1&{ - 5}&2\end{array}} \right]\)

So, the change of coordinate matrix from B to Cis

\(\mathop P\limits_{B \to C} = \left[ {\begin{array}{*{20}{c}}{ - 3}&1\\{ - 5}&2\end{array}} \right]\).