DSE 2018 Question 10: Vector spaces, rank and eigenvalues

The set $\{f_1, \dots, f_n\}$, where each $f_k$ is a real-valued function defined on $\mathbb{R}$, is said to be linearly independent if $c_1, \dots, c_n \in \mathbb{R}$ and $\sum_{k=1}^{n} c_k f_k(x) = 0$ for every $x \in \mathbb{R}$ implies $c_1 = \dots = c_n = 0$. Suppose $f_k(x) = x^k$ for all $x \in \mathbb{R}$ and $k = 1, \dots, n$. Then,