Question

Let \(V\) be a vector space with a basis \(B = \left\{ {{b_1},........{b_n}} \right\}\), \(W\) be the same vector space as \(V\), with a basis \(C = \left\{ {{c_1},........{c_n}} \right\}\) and \(I\) be the identity transformation \(I:V \to W\). Find the matrix for \(I\) relative to \(B\) and \(\xi \). What was this matrix called in section 4.7?

Short answer

The matrix \(I\) relative to \(B\) and \(C\) is the set of vectors\(M = \left( {{{\left( {{{\bf{b}}_1}} \right)}_C}\,{{\left( {\,{{\bf{b}}_2}} \right)}_C}\,\,\,\,....\,\,\,{{\left( {{{\bf{b}}_n}} \right)}_C}} \right)\). The matrix \(M = \left( {{{\left( {{{\bf{b}}_1}} \right)}_C}\,{{\left( {\,{{\bf{b}}_2}} \right)}_C}\,\,\,\,....\,\,\,{{\left( {{{\bf{b}}_n}} \right)}_C}} \right)\) is called the change of coordinates matrix from \(B\) to \(C\).

Step-by-step solution

Use the given information

It is given that \(I:V \to W\), such that, \(I\left( {\bf{x}} \right) = {\bf{x}}\). So, for each \({j^{th}}\) vector of the basis \(B\) and \(C\), the identity \(I\left( {\bf{x}} \right) = {\bf{x}}\) must be satisfied, such that \({\left( {I\left( {{{\bf{b}}_j}} \right)} \right)_C} = {\left( {{{\bf{b}}_j}} \right)_C}\). This implies that matrix \(I\) relative to \(B\) and \(C\) is the set of vectors \(M = \left( {{{\left( {{{\bf{b}}_1}} \right)}_C}\,{{\left( {\,{{\bf{b}}_2}} \right)}_C}\,\,\,\,....\,\,\,{{\left( {{{\bf{b}}_n}} \right)}_C}} \right)\).

Define the matrix 

In section 4.7, the change of coordinates matrix from \(B\) to \(C\), \(\mathop P\limits_{B \leftarrow C} \), is defined, in which columns \(\mathop P\limits_{B \leftarrow C} \) are the \(C\)-coordinate vectors of the vectors in the basis \(B\). So, \(M = \left( {{{\left( {{{\bf{b}}_1}} \right)}_C}\,{{\left( {\,{{\bf{b}}_2}} \right)}_C}\,\,\,\,....\,\,\,{{\left( {{{\bf{b}}_n}} \right)}_C}} \right)\) is called change of coordinates matrix from \(B\) to \(C\).