$F$ be a field and $K$ a subfield of $F$ ; $A \in M(n,K)$ ; $\exists (0 \ne )x \in F^n$ s.t. $Ax=0$ ; is there $0 \ne y \in K^n$ s.t. $Ay=0$?

Question

Let $F$ be a field and $K$ a subfield of $F$ ; let $A \in M(n,K)$ ; if $\exists (0 \ne )x \in F^n$ such that $Ax=0$ , then does there exist $0 \ne y \in K^n$ such that $Ay=0$ ?

Answer 1

This is immediate using determinants. If such an $x$ exists, then $\det A=0$, which means such a $y$ exists. The point is that $\det A$ detects whether $A$ has a nontrivial kernel, and the computation of $\det A$ obviously does not depend on what field you are working over.

Answer 2

Yes, if there is a nonzero $x$ over $F$ with $Ax=0$ then $A$ is singular, so there is also a nonzero $y$ over $K$ with $Ay=0$.