Relation between rank of matrix over $K$ and over $L\supset K$

Question

Let $K$ be a subfield of a larger field $L$ (e.g. $\mathbb{Q}$ over $\mathbb{R}$). Let $A$ be an $m\times n$ matrix with entries in $K$. Then $A$ can be regarded as either a matrix in $K(m,n)$ or as a matrix in $L(m,n)$. Show that the rank of $A$ is the same in either case.

(I'm thinking that the rank of $A$ is the number of non-zero rows in its row-reduced form, but I don't know how to construct a formal proof.)

Thanks!

Answer 1

The ranking of a matrix is the dimension of the vector space generated by (say) the columns. In either case the generators are in $K^n$ and they span a vector subspace of $K^n$.

So, everything boils down to $\dim_K(V)=\dim_L(V\otimes L)$ which is valid for all $K$-vector spaces $V$.