(a) Let $x_1, x_2, \ldots, x_n$ be a set of distinct sample points. Prove that the functions $f_1(x), \ldots, f_k(x)$ are linearly independent if their sample vectors $\mathbf{f}_1, \ldots, \mathbf{f}_k$ are linearly independent vectors in $\mathbb{R}^n$. (b) Give an example of linearly independent functions that have linearly dependent sample vectors. (c) Use this method to prove that the functions $1, \cos x$, $\sin x, \cos 2 x, \sin 2 x$, are linearly independent. Hint: You need at least 5 sample points.