How to prove that the $n$ functions $$ e^{r_{i}t}, te^{r_{i}t},...,t^{k_{i}-1}e^{r_{i}t} $$ $i=1,2,...,s$, where $k_{1}+k_{2}+...k_{s}=n$ and $r_{1},r_{2},...,r_{s}$ are distinct numbers , and are linearly independent on any interval $I$. My try:
Let
$a_{11}e^{r_{1}t} + a_{12}te^{r_{1}t} +...+ a_{1k_{1}}t^{k_{1}-1}e^{r_{1}t}+a_{21}e^{r_{2}t}+a_{22}te^{r_{2}t}+...+a_{21}e^{r_{2}t} +a_{2k_{2}}t^{k_{2}-1}e^{r_{2}t}+...+a_{s1}e^{r_{s}t}+a_{s2}te^{r_{s}t}+...+a_{sk_{s}}t^{k_{s}-1}e^{r_{s}t} =0. $
Then I differentiate above equation $n-1$ times to get a system of the form $AX=0$, now I need to prove $det(A) \ne 0$, but I am not getting any patterns here. Please help me to figure out this problem.