- 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.
15. $\quad$ $(-\cos x, \sin x).$
Solution. $\quad$ Given the basis $(-\cos x, \sin x)$ for $V,$ $D(V),$ and $D^2(V),$ we get the following: \begin{align*} D(-\cos x) &= \sin x \\ &= 0(\cos x) + 1(\sin x) \\ D(\sin x) &= \cos x \\ &= -1(-\cos x) + 0(\sin x) \\ \\ D^2(-\cos x) &= D(\sin x) \\ &= \cos x \\ &= -1(-\cos x) + 0 (\sin x) \\ D^2(\sin x) &= D(\cos x) \\ &= -\sin x \\ &= 0(-\cos x) - 1(\sin x) \end{align*} Which gives us the following matrices of $D$ and $D^2$ relative to the basis $(-\cos x, \sin x)$ \begin{align*} (d'_{ik}) &= \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \qquad (d''_{ik}) = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \quad \blacksquare \end{align*}