Let $n>n'\gg m$ and $V$ be a subspace of $\mathbb{C}^n$ of dimension $n'$. I am trying to characterize the set $X$ of $m\times n$ matrices $A=(a_{ij})$ satisfying $\ker(A)\supsetneq V$, that is, $A\mathbb{x}=0$ admitting a solution not in $V$.

I think $X$ is the image of a closed subscheme of $(\mathbb{P}^n\setminus \mathbb{P}^{n'})\times\mathbb{A}^{m\times n}$ (cut out by $A\mathbb{x}=0$) under the projection to the second factor $\mathbb{A}^{m\times n}$. As $n>n'\gg m$, a generic matrix $A$ lies in $X$. So $X$ is dense in $\mathbb{A}^{m\times n}$. And it is constructible by Chevalley's theorem on constructible sets (is it open in $\mathbb{A}^{m\times n}$?). I am wondering if it is possible to explicitly determine the system of polynomial equations cuting out the closure of $\mathbb{A}^{m\times n}\setminus X$. For simplicity just assume the subspace $V$ is spanned by the first $n'$ coordinate vectors, if it helps.

Thanks!

Let $n>n'\gg m$ and $V$ be a subspace of $\mathbb{C}^n$ of dimension $n'$. I am trying to characterize the set $X$ of $m\times n$ matrices $A=(a_{ij})$ satisfying $\ker(A)\not\subseteq V$, that is, $A\mathbb{x}=0$ admitting a solution not in $V$.

I think $X$ is the image of a closed subscheme of $(\mathbb{P}^n\setminus \mathbb{P}^{n'})\times\mathbb{A}^{m\times n}$ (cut out by $A\mathbb{x}=0$) under the projection to the second factor $\mathbb{A}^{m\times n}$. As $n>n'\gg m$, a generic matrix $A$ lies in $X$. So $X$ is dense in $\mathbb{A}^{m\times n}$. And it is constructible by Chevalley's theorem on constructible sets (is it open in $\mathbb{A}^{m\times n}$?). I am wondering if it is possible to explicitly determine the system of polynomial equations cuting out the closure of $\mathbb{A}^{m\times n}\setminus X$. For simplicity just assume the subspace $V$ is spanned by the first $n'$ coordinate vectors, if it helps.

Thanks!