# 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.