Does multiplying by $i=\sqrt{-1}$ count as multiplying by a constant for the purposes of linear independence? E.g. are $i \sin(x)$ and $\sin(x)$ linearly independent or linearly dependent? 
For context, I'm trying to reconcile the $e^{ix} = \cos(x) + i\sin(x)$ formula with the generalized solution to a homogeneous DE, $y=C_1e^{ax}\cos(bx) +C_2e^{ax}\sin(bx)$ and I think I'm just getting a bit confused.
 A: Depends on the ground field: $\sin(x)$ and $i\sin(x)$ are:


*

*linearly dependent over $\Bbb C$;

*linearly independent over $\Bbb R$
A: To even make sense of the formula
$$e^{ix}=\cos(x)+i\sin(x)$$
you are fundamentally using complex numbers. Therefore constants are "any complex number", of which $i$ is one.
So when treating them as elements of a vector space of functions, $\sin(x)$ and $i\sin(x)$ are linearly dependent.
Note that you're running specific linear combinations
$$\cos(x)=\frac{e^{ix}+e^{-ix}}{2}\text{,}\quad\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$
to recover a purely real function. This does not stop the fact that:


*

*$e^{ix}$ and $e^{-ix}$ are two linearly independent functions

*Therefore you can recover two linearly independent functions by a change of basis

*Even though these functions happen to be purely real for purely real arguments.



As the answer by Fabio Lucchini states, if you're considering functions $f:\mathbb{R}\rightarrow\mathbb{C}$ as a real vector space, then they are linearly independent.
However, this interpretation does not make sense. A vector space is where you can, for example, multiply elements by scalars. A function space $f:A\rightarrow B$ is, by the cardinal exponentiation interpretation, "a list of elements of $B$ indexed by elements of $A$". This means, for example,
$$(cf)(x)=c(f(x))$$
as an axiom. This multiplication is between $c$ and $f(x)\in B$. Therefore, using any scalar field other than $c\in B$ (in this case $\mathbb{C}$) doesn't work.
