Column rank/Product of matrices/Estimate/Exercise

Let ${}B$ be an ${}n\times p$-matrix, and let ${}A$ be an ${}m\times n$-matrix. Show that, for the column rank, the estimate

${}\operatorname {rk} \,A\circ B\leq {\min {\left(\operatorname {rk} \,A,\operatorname {rk} \,B,\right)}}\,$

holds.

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