I have this problem:
Let $c$ be a Frenet curve in $\mathbb{R}^n$. Show that $ \operatorname{Det}\left(c^{\prime}, c^{\prime \prime}, \ldots, c^{(n)}\right)=\prod_{i=1}^{n-1}\left(\kappa_i\right)^{n-i} $
I've been struggling trying to understand how to solve it. This problem is from Kuhnel's Differential Geometry book.
I have these two properties that the book gives, I think are involved in the problem:
2.4. Definition. (Frenet curve) Let $c(s)$ be a regular curve in $\mathbb{R}^n$, which is parametrized by arc length and $n$-times continuously differentiable. Then $c$ is called a Frenet curve, if at every point the vectors $c^{\prime}, c^{\prime \prime}, \ldots, c^{(n-1)}$ are linearly independent. The Frenet $n$-frame $e_1, e_2, \ldots, e_n$ is then uniquely determined by the following conditions: (i) $e_1, \ldots, e_n$ are orthonormal and positively oriented. (ii) For every $k=1, \ldots, n-1$ one has $\operatorname{Lin}\left(e_1, \ldots, e_k\right)=$ $\operatorname{Lin}\left(c^{\prime}, c^{\prime \prime}, \ldots, c^{(k)}\right)$, where $\operatorname{Lin}$ denotes the linear span. (iii) $\left\langle c^{(k)}, e_k\right\rangle>0$ for $k=1, \ldots, n-1$.
and
2.13. Theorem and definition. (Frenet equations in $\mathbb{R}^n$ ) Let $c$ be a Frenet curve in $\mathbb{R}^n$ with Frenet $n$-frame $e_1 \ldots, e_n$. Then there are functions $\kappa_1, \ldots, \kappa_{n-1}$ defined on that curve with $\kappa_1, \ldots, \kappa_{n-2}>0$, so that every $\kappa_i$ is $(n-1-i)$-times continuously differentiable and $$ \left(\begin{array}{c} e_1 \\ e_2 \\ \vdots \\ \vdots \\ \vdots \\ e_{n-1} \\ e_n \end{array}\right)^{\prime}=\left(\begin{array}{cccccc} 0 & \kappa_1 & 0 & 0 & \cdots & 0 \\ -\kappa_1 & 0 & \kappa_2 & 0 & \ddots & \vdots \\ 0 & -\kappa_2 & 0 & \ddots & \ddots & \vdots \\ 0 & 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & 0 & \kappa_{n-1} \\ 0 & \cdots & \cdots & 0 & -\kappa_{n-1} & 0 \end{array}\right)\left(\begin{array}{c} e_1 \\ e_2 \\ \vdots \\ \vdots \\ \vdots \\ e_{n-1} \\ e_n \end{array}\right) . $$ $\kappa_i$ is called the $i$-th Frenet curvature and the equations are called the Frenet equations.
So my first attempt I thought was an induction over $n$, but it didn't work. So I was thinking in something like QR decomposition, but I'm not sure if it could work. Do you have any suggestion to solve this problem?