Suppose that there are four matirces $A$, $B$, $C$ and $D$, and assume that $[A,B,C,D]$ is of full column rank. Then, I want to know the rank of $[A \otimes B, C \otimes D]$.
Alternatively, suppose that there are three vectors $a,b$ and $c$ and assume that $[a,b,c]$ is of full column rank with rank 3. Then, it seems that $[a \otimes a, b \otimes b]$ has column rank 2 and $[a \otimes a, b \otimes b, c \otimes c]$ has column rank 3. But, it is not trivial for me to prove this.
Any help would be much appreciated!