Nullity of the same matrix over a finite and infinite field Is there a relation between the nullity of the same matrix over a finite field and a infinite field?
Or is there a prime power q so there the differents of nullity of a matrix over, let's say the rational numbers and the field $\Bbb{F_q}$ is small?

Let $p$ be a prime; then the nullity of
\begin{bmatrix}
p & 2p \\
p & 3p
\end{bmatrix}
is zero over $\mathbb{Q}$ (and every infinite field of characteristic $0$); on the other hand, the matrix has nullity two over the $p$-element field (and every extension thereof).
If $K$ is an extension of the field $F$, then the rank and nullity of a matrix over $F$ don't change if we consider it as a matrix over $K$, because the rank can be computed with Gaussian elimination and is the same over $F$ and over $K$.
