Given $f_1, f_2, \ldots, f_n \in C^{n-1}(\mathbb{R})$. Prove that if there is $v$ such that the matrix
$$
\left( \begin{array}{ccc}{f_{1}(v)} & {\ldots} & {f_{n}(v)} \\ {f_{1}^{\prime}(v)} & {\cdots} & {f_{n}^{\prime}(v)} \\ {\vdots} & {\ddots} & {\vdots} \\ {f_{1}^{(n-1)}(v)} & {\dots} & {f_{n}^{(n-1)}(v)}\end{array}\right)
$$
is invertible, then, $f_1, f_2, \ldots, f_n$ are linearly independent.
I can't do it.
Best Answer
If $ \sum\limits_{k=1}^{n} c_kf_k=0$ then $ \sum\limits_{k=1}^{n} c_kf_k^{(j)}=0$ for $0 \leq j \leq n-1$. In particular $ \sum\limits_{k=1}^{n} c_kf_k^{(j)}(v)=0$ for $0 \leq j \leq n-1$. Since the matrix of this linear system is invertible we get $c_i=0$ for all $i$.