A problem on rank of a matrix over two fields This is a problem from Berkeley problems in mathematics.
If $F$ is a subfield of $K$, and $M$ has entries in $F$, how is the row rank of $M$ over $F$ related to the row rank of $M$ over $K$?
where $M$ is a n by n matrix
The solution says "If a set of rows of $M$ is linearly independent over $F$, then clearly it is also independent over K, so the rank of $M$ over $F$ is, at most, the rank of $M$ over $K$."
I have some trouble understanding this, what I thought was that if they are linearly independent over the bigger field K, they are linearly independent over F. (Because all linear combinations with scalars from F are subsumed when you are talking about linear combinations in K) However here it is the other way around

Doesn't necessarily need to be the case always. Consider the matrix
$$ A =
\begin{bmatrix}
    1       & 1  \\
    i       & i  \\
\end{bmatrix}
$$
$A$ has rank $2$ over $\Bbb{R}$ but $1$ over $\Bbb{C}$.
If $A$ is a matrix of rank $r$ over a field $F$ then it has rank $r$  over any extension field of $F$. But if $A$ is a matrix of rank $r$ over a field $K$ then its rank over any sub-field of $K$ is at least $r$. But it can be more also.
