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 $0$.