4.4.1 change of bases

Example

• Let $B=((1,1),(1,-1))$ and $C=((1,2),(1,3))$ find the change of basis matrix $,P_{B\to C}$

$$\begin{array}{r}[{\vec{v}}_1{]}_c=?\\ \left[\begin{array}{ccc}1& 1& |1\\ 2& 3& |1\end{array}\right]\\ [{\vec{v}}_2]=?\\ \left[\begin{array}{ccc}1& 1& |1\\ 2& 3& |-1\end{array}\right]\\ \left[\begin{array}{cccc}1& 1& |1& 1\\ 2& 3& |1& -1\end{array}\right]⟹\left[\begin{array}{cccc}1& 0& |2& 4\\ 0& 1& |-1& -3\end{array}\right]\\ P_{B\to C}=\left[\begin{array}{cc}2& 4\\ -1& -3\end{array}\right]\end{array}$$

• We just find each matrix and solve for the coordinates of the original ones

• The determinant of a change of basis matrix cannot be 0