Exercise 5.2.15

Following the step of the previous exercise, we may pick a matrix $Q$ whose column vectors consist of eigenvectors and $Q$ is invertible. Let $D$ be the diagonal matrix $Q^{-1}\mathit{AQ}$. And we also know that finally we’ll have the solution $x=\mathit{Qu}$ for some vector $u$ whose $i$-th entry is $c_i{e}^{\lambda }$ if the $i$-th column of $Q$ is an eigenvector corresponding to $\lambda$. By denoting $\overline{D}$ to be the diagonal matrix with ${\overline{D}}_{\mathit{ii}}=e^{D_{\mathit{ii}}}$, we may write $x=Q\overline{D}y$. where the $i$-th entry of $y$ is $c_i$. So the solution must be of the form described in the exercise.

For the second statement, we should know first that the set

are linearly independent in the space of real functions. Since $Q$ invertible, we know that the solution set

is an $n$-dimensional real vector space.