how to show that $\{ \cos(k\omega x): 1\leq k \leq n, n\in\mathbb N \}$ is L.I? I need show that $\cos(\omega x),\cos(2\omega x), \cos(3\omega x), \ldots,\cos(n\omega x) $ is linearly independent? i'm studing linear algebra for physics and I can't use an inner product, I try to show this using the Wronskian but it seems so difficult. How can I do it? 

They're eigenfunctions of $D=\frac{d^{2}}{dx^{2}}$ with different eigenvalues:
$$
             D\cos(k\omega x)=-k^{2}\omega^{2}\cos(k\omega x).
$$
Therefore, if
$$
              \sum_{k}A_k\cos(k\omega x) =0,
$$
you can show that $A_n=0$ for every $n$ because
$$
     0=\prod_{k\ne n}(D+k^{2}\omega^{2})\sum_k A_k\cos(k\omega x)=\prod_{k\ne n}(-n^{2}\omega^{2}+k^{2}\omega^{2})A_n\cos(n\omega x).
$$
