- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
2.12 Exercises
$\quad$ In the linear space of all real-valued functions, each of the following sets is independent and spans a finite-dimensional subspace $V.$ Use the given set as a basis for $V$ and let $D: V \rightarrow V$ be the differentiation operator. In each case, find the matrix of $D$ and of $D^2$ relative to this choice of basis.
17. $\quad$ $(e^x \sin x, e^x \cos x).$
Solution. $\quad$ Given the basis $(e^x \sin x, e^x \cos x)$ for $V,$ $D(V),$ and $D^2(V),$ we get the following: \begin{align*} D(e^x \sin x) &= e^x \sin x + e^x \cos x \\ D(e^x \cos x) &= -e^x\sin x + e^x \cos x \\ \\ D^2(e^x \sin x) &= D(e^x \sin x + e^x \cos x) \\ &= D(e^x \sin x) + D(e^x \cos x) \\ &= (e^x \sin x + e^x \cos x) + (-e^x \sin x + e^x \cos x) \\ &= 0(e^x \sin x) + 2(e^x \cos x) \\ D^2(e^x \cos x) &= D(-e^x\sin x + e^x \cos x) \\ &= D(e^x \cos x) - D(e^x \sin x) \\ &= (-e^x\sin x + e^x \cos x) - (e^x \sin x + e^x \cos x) \\ &= -2(e^x \sin x) + 0 (e^x \cos x) \end{align*} Which gives us the following matrices of $D$ and $D^2$ relative to the basis $(e^x \sin x, e^x \cos x)$ \begin{align*} (d'_{ik}) &= \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} \qquad (d''_{ik}) = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} \quad \blacksquare \end{align*}