Question

Let \(V\) be \({\mathbb{R}^n}\) with a basis \(B = \left\{ {{b_1},........{b_n}} \right\}\); let \(W\) be \({\mathbb{R}^n}\) with the standard basis, denoted here by \(\xi \); and consider the identity transformation \(I:V \to W\) , where \(I\left( {\rm{x}} \right) = {\rm{x}}\). Find the matrix for \(I\) relative to \(B\) and \(\xi \). What was this matrix called in section 4.4?

Short answer

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

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 \(\xi \), the identity \(I\left( {\bf{x}} \right) = {\bf{x}}\) must be satisfied, such that \({\left( {I\left( {{{\bf{b}}_j}} \right)} \right)_\xi } = {{\bf{b}}_j}\). This implies that matrix \(I\) relative to \(B\)and \(\xi \) is the set of vectors \(\left( {{{\bf{b}}_1}\,\,\,{{\bf{b}}_2}........\,{{\bf{b}}_n}} \right)\).

Define the matrix

In section 4.4, the change of coordinates matrix is defined, which changes the \(B\)-coordinates of a vector \(x\) into the standard coordinates for \(x\), where the change is carried out in \({\mathbb{R}^n}\)for the basis \(B = \left( {{{\bf{b}}_1}\,\,\,{{\bf{b}}_2}........\,{{\bf{b}}_n}} \right)\). So, \(\left( {{{\bf{b}}_1}\,\,\,{{\bf{b}}_2}........\,{{\bf{b}}_n}} \right)\) is called the change of coordinates matrix.