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