I am attempting to prove the following statement: if $\phi$ is a bilinear map which takes $V_1 \times V_2$ to $W$ where $V_1$ and $V_2$ are vector spaces of dimension $l_1$ and $l_2$ respectively and $\phi(v_1,v_2) \in W$ is non-zero for every $v_1 \in V_1$ and $v_2 \in V_2$, then the image of $\phi$ spans a subspace of $W$ with dimension greater than $l_1 - l_2 -1$.
My idea to prove this is to consider that tensors of rank $1$ $\{ v_1 \otimes v_2 \}$ form a subvariety of dimension $l_1 + l_2 - 1$ in $V_1 \otimes V_2$, and then the kernel of $\phi: V_1 \otimes V_2 \rightarrow W$ only intersects this subvariety at $0$.