How to prove that the set $\{\sin(x),\sin(2x),...,\sin(mx)\}$ is linearly independent? [closed]

Could you help me to show that the functions $\sin(x),\sin(2x),...,\sin(mx)\in V$ are linearly independent, where $V$ is the space of real functions?

Thanks.

Suppose that, for every $x\in\Bbb R$ we have $$a_1\sin x+a_2\sin 2x+\cdots+a_m\sin mx=0$$

Take $i\in \{1,\dots,m\}$, and consider $\sin ix$. Multiply throughout and integrate from $x=0$ to $x=2\pi$. Do this for $i=1,\dots,m$. Use that $$\int_0^{2\pi} \sin mx\sin nxdx=\begin{cases}0& m\neq n\\ \pi &m=n\end{cases}$$

ADD If the above wasn't entirely clear, for each $1\leq k\leq m$

$$\begin{align}\sum_{j=1}^m a_j\sin jx&=0\\ \sum_{j=1}^m a_j\sin kx\sin jx&=0\\ \sum_{j=1}^m a_j\int_0^{2\pi}\sin kx\sin jxdx&=0\\ a_k \pi&=0\\ {}&{}\\ a_k&=0\end{align}$$

since, of course, $\pi\neq 0$.

For variety....

Let $D$ be the differentiation operator. It is a linear transformation.

$D^2 \sin(kx) = -k^2 \sin(kx)$, so $\sin(kx)$ is an eigenfunction (i.e. an eigenvector when the vector space is a space of functions) of $D^2$ with eigenvalue $-k^2$.

Since each of the eigenvalues $-1, -4, -9, \cdots, -m^2$ are distinct, the eigenfunctions must be linearly independent.