Rank of a matrix over field extension.

Question

Let $A$ be a matrix over a field $\mathbb{F}$ and $\mathbb{K}$ be a field extension of $\mathbb{F}$. As I know that characteristic and minimal polynomial of $A$ over $\mathbb{F}$ and $\mathbb{K}$ are same. Now can I say that rank of the matrix over both field will be same? I tried with particular matrices and got the same rank. Please suggest. Thanks.

Answer 1

Suppose that $A$ is a $n\times n$-matrix which represents a linear operator, $F^n=\operatorname{Ker} f\oplus V$ where the restriction of $A$ to $V$ is an isomorphism onto $\operatorname{Im}A$. Over $K$, you have $K^n=\operatorname{Ker}f\otimes_FK\oplus V\otimes_FK$ which shows that the rank does not change with base change.