Change of Coordinates - Illustration

To illustrate the essence of ‘change of coordinates’, consider two ordered bases \(\EuScript{A} = (\Vect{a}_1,\Vect{a}_2)\) and \(\EuScript{B}=(\Vect{b}_1,\Vect{b}_2)\) in \(\RNr[2]\). Now, given a vector \(\Vect{x}\) in \(\RNr[2]\), we may

1. express \(\Vect{x}\) in a unique way as a linear combination of \(\EuScript{A}:\)

   \[\Vect{x} = u \Vect{a}_1 + v \Vect{a}_2,\quad \text{and so}\quad \Vect{x}_{\EuScript{A}} = (u,v)\]

   The \(\EuScript{A}\)-coordinates of \(\Vect{x}\) are \((u,v)\)

2. express \(\Vect{x}\) in a unique way as a linear combination of \(\EuScript{B}:\)

   \[\Vect{x} = s \Vect{b}_1 + t \Vect{a}_2,\quad \text{and so}\quad \Vect{x}_{\EuScript{B}} = (s,t)\]

   The \(\EuScript{B}\)-coordinates of \(\Vect{x}\) are \((s,t)\)

Here \(\Vect{x}_{\EuScript{A}} = \left( \tfrac{3}{2} , \tfrac{5}{2} \right)\)

So we arrive at the question: suppose we know the \(\EuScript{A}\)-coordinates of \(\Vect{x}\), what are its \(\EuScript{B}\)-coordinates? and, vice versa, suppose we know the \(\EuScript{B}\)-coordinates of \(\Vect{x}\), what are its \(\EuScript{A}\)-coordinates?

In the section on ‘Change of Coordinates’, we learn that there is a unique matrix \(\Mtrx{C}_{\EuScript{B}\EuScript{A}}\), here of size \((2,2)\), with

\[\Vect{x}_{\EuScript{B}}\ =\ \Mtrx{C}_{\EuScript{B}\EuScript{A}}\cdot \Vect{x}_{\EuScript{A}}\]

Moreover, the matrix which reverses the change from \(\EuScript{A}\)-coordinates to \(\EuScript{B}\)-coordinates is

\[\Mtrx{C}_{\EuScript{A}\EuScript{B}}\ =\ \Mtrx{C}_{\EuScript{B}\EuScript{A}}^{-1}\]