Show linear independence of cosines We're given a subset of the vector space of continuous real functions on $\mathbb{R}$ defined as
$$
\{\cos(cx) \;\lvert\; c \in \mathbb{R},\; c > 0\}
$$
How do we show this set is linearly independent? 
I saw some other questions like this on the site, but all of them used $\cos(x)$ with no constant multiplying $x$. 

Let $V$ denote the vector space of infinitely-differentiable functions on $\mathbf{R}$. Define a linear operator $T$ on $V$ by $$Tf = f''.$$
For $c > 0$, let $f_c$ be the function defined by
$$f_c(x) = \cos(cx).$$
Then
$$Tf_c = -c^2 f_c.$$
In other words, $f_c$ is an eigenvector of $T$ with eigenvalue $-c^2$. The (easy) theorem that a collection of eigenvectors corresponding to distinct eigenvalues is linearly independent now implies that $\{f_c: c >0\}$ is linearly independent.
