
linearly independent
Is the set $\displaystyle \{ \cos(nx) ,\sin(mx) \mid m,n \in \mathbb{N} \} \subseteq \{ f:\mathbb{R} \rightarrow \mathbb{R}\} $ linearly independent?
#I have a vague kind of idea that those functions form a linearly independent set but only on [0,2pi]. Could anyone give me some advice on this? Thanks very much

Yes, they form a linearly independent set on the entire real line, not just $\displaystyle [0, 2\pi]$. Indeed, because they are periodic with period $\displaystyle 2\pi$, whatever is true of them on $\displaystyle [0, 2\pi]$ is true for all x.

Thanks! is there a way to prove that those functions are linearly independent in a simple way? It's easy to show if i take 2 functions out of it
ie $\displaystyle \alpha \sin (nx) + \beta \cos (mx) = 0 \implies \alpha = \beta = 0 $ but I am having trouble doing it for an arbitrary subset