Let $K/F$ be a finite field extension and let $A$ and $B$ be finite-dimensional $K$ vector spaces. Let $\phi$ be the $F$-linear map $\phi : A \otimes_{F} B \to A \otimes_{K} B$ taking $a\otimes_{F} b$ to $a\otimes_{K} b$.
Claim: The kernel of $\phi$ is the $F$ vector subspace $I$ generated by all elements $ca\otimes_{F} b - a\otimes_{F} cb$ with $a\in A$, $b \in B$, and $c\in K$.
I can see that $I$ is contained in the kernel of $\phi$ but why must the converse hold?