Question 4.3.5: Let V be the subspace of C^β spanned by the functions e^x an......
Let V be the subspace of C^β spanned by the functions e^x and e^{βx} , with the bases π = (e^x , e^{βx} ) and π = (e^x + e^{βx} , e^x β e^{βx} ). Find the change of basis matrix S_{π βπ}.
By Definition 4.3.3,
S=\begin{bmatrix}[e^x + e^{βx}]_π&[e^x β e^{βx}]_π\end{bmatrix}.
Now
It is suggestive to write the functions ex and e^{βx} of basis π next to the rows of matrix S_{π βπ}, while the functions e^x + e^{βx} and e^x β e^{βx} of basis π are written above the columns:
e^x + e^{βx} \ \ \ \ \ \ e^x – e^{βx}
S_{ \mathfrak{B}\rightarrow \mathfrak{A} }=\left [ \begin{matrix} &1&&&&1&\\&1&&&&-1& \end{matrix} \right ] \begin{matrix} \ \ \ e^xΒ \\ \ \ \ e^{-x} \end{matrix}
The second column of matrix S indicates that e^x β e^{β1} = 1 Β· e^x + (β1) Β· e^{βx}.