# Linear (in)dependence of minors of a matrix

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?

Consider the map $\phi: V\to K^{M}$ where $M=\binom{n+1}{r+1}$, which sends a vector $y\in V$ to the $M-$tuple of minors (ordered as you wish) of the matrix $$A_y=(y\ \vert\ v_1\ \vert\;\cdots\;\vert\ v_{r+1})$$ then $y\in W$ if and only if $\phi(y)=0$, because $y\in W$ iff it is linearly dependent on $\{v_1,\ldots, v_{r+1}\}$, fact that happens iff $\mathrm{rk}A_y=r+1$ iff all the $(r+2)$-minors of $A_y$ vanish. Therefore, $W=\ker \phi$. Now, write $\phi=(\phi_1,\ldots, \phi_M)$, with $\phi_j\in V^*$. We have that $W=\{\phi_1=\ldots=\phi_M=0\}=\left(\mathrm{Span}\{\phi_1,\ldots,\phi_M\}\right)^0$, but $\dim W=r+1$, so $\dim\mathrm{Span}\{\phi_1,\ldots,\phi_M\}=(n+1)-(r+1)=n-r$.