Showing that exponential functions are linearly independent

  • #1
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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.

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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?
 

Answers and Replies

  • #2
andrewkirk
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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##.
 
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