Proof that this is independent Prove that {$1, \sin(x), \sin(2x), \sin(3x),\ldots, \sin(nx)$} is an independent set.
I can think of the long way which is to differentiate this and put the differentiations into a matrix and row reduce it.
Please advice, thanks!

Hint: Suppose that for some real numbers $k, \alpha_1, \alpha_2, \ldots \alpha_n$ we have
$$ k = \sum_{i=1}^n \alpha_i \sin (ix). $$
Then, multiply by $\sin(jx)$ and integrate from $0$ to $2 \pi$ to conclude that $\alpha_j = 0$. Hence $k=0$.
