Question 4.3.8: As in Example 5, let V be the linear space spanned by the fu......
As in Example 5, let V be the linear space spanned by the functions e^x and e^{−x} , with the bases 𝔄 = (e^x , e^{−x} ) and 𝔅 = (e^x + e^{−x} , e^x − e^{−x} ). Consider the linear transformation D( f ) = f^{\prime} from V to V.
a. Find the 𝔄-matrix A of D.
b. Use part (a), Theorem 4.3.5, and Example 5 to find the 𝔅-matrix B of D.
c. Use Theorem 4.3.2 to find the 𝔅-matrix B of D in terms of its columns.
a. Let’s use a diagram. Recall that (e^{−x} )^{\prime} = −e^{−x} , by the chain rule.
b. In Example 5 we found the change of basis matrix S =\begin{bmatrix}1&1\\1&−1\end{bmatrix} from 𝔅 to 𝔄. Now
B=S^{-1}AS=\frac{1}{2}\begin{bmatrix}1&1\\1&-1\end{bmatrix}\begin{bmatrix}1&0\\0&-1\end{bmatrix}\begin{bmatrix}1&1\\1&-1\end{bmatrix}=\begin{bmatrix}0&1\\1&0\end{bmatrix}.
c. Note that D(e^x + e^{−x} ) = e^x − e^{−x} and D(e^x − e^{−x} ) = e^x + e^{−x} . Thus,
D(e^x + e^{−x} ) \ \ \ \ D(e^x – e^{−x} )
B=\left [ \begin{matrix} &0&&&&1&\\&1&&&&0& \end{matrix} \right ] \begin{matrix} \ \ \ e^x +e^{-x} \\ \ \ \ e^x -e^{-x} \end{matrix}