Let $\displaystyle x_1 , x_2, ..., x_k$ be vectors in $\displaystyle \mathbb{R}^n$; let $\displaystyle X$ be a matrix $\displaystyle X = [x_1 , x_2, ..., x_k]$. If $\displaystyle I = (i_1, i_2,...,i_k)$ is an arbitrary $\displaystyle k$-tuple from the set $\displaystyle \{1, 2, ... ,n\}$, show that

$\displaystyle \phi_{i_1} \wedge \phi_{i_2} \wedge ... \wedge \phi_{i_k} (x_1 , x_2, ..., x_k) = det X_I$,

where

$\displaystyle \phi_{i}(e_j) = \begin{cases} 1, & \mbox{if } i=j \\ 0, & \mbox{if } i \neq j \end{cases}$

for $\displaystyle i, j \in \{1,2,...,n\}.$

Please help or give a hint