# linearly independent

• Oct 13th 2010, 04:51 PM
slevvio
linearly independent
Is the set $\{ \cos(nx) ,\sin(mx) \mid m,n \in \mathbb{N} \} \subseteq \{ f:\mathbb{R} \rightarrow \mathbb{R}\}$ 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
• Oct 14th 2010, 05:02 AM
HallsofIvy
Yes, they form a linearly independent set on the entire real line, not just $[0, 2\pi]$. Indeed, because they are periodic with period $2\pi$, whatever is true of them on $[0, 2\pi]$ is true for all x.
• Oct 14th 2010, 08:09 AM
slevvio
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 $\alpha \sin (nx) + \beta \cos (mx) = 0 \implies \alpha = \beta = 0$ but I am having trouble doing it for an arbitrary subset