Can the coefficients used to prove a set of functions is linearly dependent be imaginary?

Question

Example: $\cos x$, $e^{ix}$, $3\sin x$.

I can show: $C_1\cos x + C_2 e^{ix} + C_33\sin x = 0$ if $(C_1,C_2,C_3) = (1,-1,i/3)$

But i don't know if $C_3 = i/3$ is a valid coefficient to choose. Can such coefficients be imaginary?

Answer 1

Linear dependency also depends on the field

For example take $\mathbb{F}=\mathbb{Q}$ then we can prove that $\sqrt{2},\sqrt{3}$ are linearly independent over $\mathbb{Q}$, but clearly they are not linearly independent in $\mathbb{F}=\mathbb{R}$

The underlying field should be mentioned in the question asked