Showing that exponential functions are linearly independent

Mr Davis 97

1. Homework Statement
If $r_1, r_2, r_3$ are distinct real numbers, show that $e^{r_1t}, e^{r_2t}, e^{r_3t}$ are linearly independent.

2. Homework Equations

3. The Attempt at a Solution
By book starts off by assuming that the functions are linearly dependent, towards contradiction. So $c_1e^{r_1t} + c_2e^{r_2t}+ c_2e^{r_3t} = 0$. After differentiating and doing some manipulations, the book finds that $e^{(r_1 - r_2)t} = C e^{(r_3 - r_2)t}$, where C is just some constant. It then states that this is a contradiction, so the original statement must be true. I am, however, a little confused as to why this is a contradiction. Is it a contradiction based on some previously shown result that two exponential functions with different powers can never be linearly dependent?

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andrewkirk

Homework Helper
Gold Member
We need a few more steps to get a formal contradiction:

If we multiply both sides by $e^{(r_2-r_3)t}$ the equation becomes $e^{(r_1-r_3)t}=C$. Substituting successively 0 and 1 for $t$ we get
$$1=e^{(r_1-r_3)\cdot 0}=C=e^{(r_1-r_3)\cdot 1}=e^{r_1-r_3}\neq 1$$
where the last inequality follows from the fact that $r_1\neq r_3$.

"Showing that exponential functions are linearly independent"

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