# Which of these functions are linearly independent?

It looks to me like they are all linearly independent but that is coming back as incorrect.

What is the general procedure for doing this anyway, I presume it is setting $c_1f(x) + c_2g(x) = 0$ and trying to find out if there are any non trivial solutions for c1 and c2...that is what I did but in every case I found c1 and c2 to be zero.

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## 2 Answers

For example the second case : they are linear dependent.Because one is equal to e^-9 multipliying the other for all x.

First case and the last : in the interval x>0 they are linear dependent, also in the interval with x<0, but there is no constant for all R, so in R they are linear independent. The third case are independent. You can suppose that one is equal to the other multiplied by a constant and get a contradiction. Or go to the definition, substitute two values of x, form a system , find that the only scalars are 0, and then they are the unique scalars for all x.

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All but 2. are linearly independent. The second pair of functions is linearly dependent, as $$g(x)=e^{3(x-3)}=e^{3x-9}=e^{-9}e^{3x}=e^{-9}f(x)$$ so $e^{-9}f(x)+(-1)g(x)=0$.

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