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Supposing $\displaystyle A\in \mathbb{M}_{np}(\mathbb{R})$ and $B\in\mathbb{M}_{pq}(\mathbb{R})$:

How can prove that: $\displaystyle rang(AB) \le \inf(rang(A),rang(B))$

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The idea is to note that if $V$ is any subspace of $\mathbb{R}^p$, then the dimension of $W=\{Av:v\in V\}$ is at most $\dim(V)$. This follows from applying the Rank-Nullity Theorem to the restricted linear transformation $T:V\rightarrow\mathbb{R}^n$, $T(v)=Av$. Setting $V=range(B)$ and noting that $range(AB)$ then equals $W$, we get $rank(AB)\leq rank(B)$.

Since $range(AB)$ is a subspace of $range(A)$, we also have that $rank(AB)\leq rank(A)$.

Our findings give $rank(AB)\leq min(rank(A),rank(B))$.

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