From (Italian, very nice book):"Lezioni di Geometria Analitica e Proiettiva" by Beltrametti, CArletti, Gallarati, Bragadin (pag. 21):


Let $K$ a field, $V:= K^{n+1}$ and let  $e_1,\ldots, e_{n+1}$ a base (canonical or not) of $V$. Let $W\subset K^{n+1}$ a $K$-vectorial subspace with dimension $r+1$, and let $v_1,\ldots, v_{r+1}$ a base of $W$, with 

$v_m= a^1_m\cdot e_1 + \ldots a^{n+1}_m\cdot e_{n+1}$ for $1\leq m\leq r+1$


Let $M$ the  matrix with  ($n+1$) row's:



$x_1, a^1_1\ldots, a^1_{r+1} $

$x_2, a^2_1\ldots, a^2_{r+1} $

$\ldots, \ldots, \ldots$

$\ldots, \ldots, \ldots$

$x_{n+1}, a^{n+1}_1, \ldots a^{n+1}$ 

(the last element is $a^{n+1}_{r+1}$)


The book assert (mentioning Kronecker theorem)  that

  *the     $r+2$-minor's of $M$* (these are $\binom{n+1}{r+2}$)


 *considered as linear forms   (grade  1 homogeneous polynomial) on  variables* $x_1,\ldots, x_{n+1}$

 *are  linearly dependent, and there are $n-r$ (and no more) linearly independent $r+2$-minors.*

Is this true?

How to prove this?