Rank bound for a matrix product
= Rank bound for a matrix product
{title2=$\operatorname{rank}(AB)\leq\min(\operatorname{rank}A,\operatorname{rank}B)$}
For composable <linear maps>, $\operatorname{im}(AB)=A(\operatorname{im}B)$ lies in $\operatorname{im}A$ and has <dimension> at most that of $\operatorname{im}B$. This proves the bound on the <rank of a matrix> product. When $A$ is $m\times p$, $B$ is $p\times n$, and $m,n\geq p$, the bound $p$ is attained by $A=\binom{I_p}{0}$ and $B=(I_p\ 0)$.