Let $\displaystyle f,g,\in{F(R,R)} $ be the functions defined by $\displaystyle f(t)=e^{rt} $and $\displaystyle g(t)=e^{st}$, where $\displaystyle r\neq{s}$. Prove that $\displaystyle f $and $\displaystyle g $are linearly independent in $\displaystyle F(R,R)$.

Assume $\displaystyle ae^{rt}+ be^{st}=0$

$\displaystyle ae^{rt} = -be^{st}$

$\displaystyle ln(a)+rt = ln(-b) +st$

$\displaystyle ln(\frac{-a}{b})=t(s-r)$

For our assumption to be true, the values of $\displaystyle a $ and $\displaystyle b $ are dependent on the value of $\displaystyle t$, therefore there is no universal value of $\displaystyle a$ or $\displaystyle b$, except $\displaystyle 0$, for which $\displaystyle af+bg=0$

Thus, these functions are linearly independent

This proof is correct?