Exercise 2.7.10

Use induction on the number $n$ of distinct scalar $c_i$’s. When $n=1$, the set $\{e^{c_{1}t}\}$ is independent since $e^{c_{1}t}$ is not identically zero. Suppose now the set $\{e^{c_{1}t},e^{c_{2}t},\dots ,e^{c_{n}t}\}$ is independent for all $n<k$ and for distinct $c_i$’s. Assume that

Since any differential operator is linear, we have

This means that $(c_i-c_k)b_i=0$ and so $b_i=0$ for all $i<k$ by the fact that $c_i$’s are all distinct. Finally $b_k$ is also zero since